chapter 12 resource masters - ktl math classes . . . . . . . . . . . . . . . . . . . . . .a2–a39...

Chapter 12Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 12 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828015-X Algebra 2Chapter 12 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 12-1Study Guide and Intervention . . . . . . . . 699–700Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 701Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 702Reading to Learn Mathematics . . . . . . . . . . 703Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 704









Chapter 12 AssessmentChapter 12 Test, Form 1 . . . . . . . . . . . 753–754Chapter 12 Test, Form 2A . . . . . . . . . . 755–756Chapter 12 Test, Form 2B . . . . . . . . . . 757–758Chapter 12 Test, Form 2C . . . . . . . . . . 759–760Chapter 12 Test, Form 2D . . . . . . . . . . 761–762Chapter 12 Test, Form 3 . . . . . . . . . . . 763–764Chapter 12 Open-Ended Assessment . . . . . 765Chapter 12 Vocabulary Test/Review . . . . . . 766Chapter 12 Quizzes 1 & 2 . . . . . . . . . . . . . . 767Chapter 12 Quizzes 3 & 4 . . . . . . . . . . . . . . 768Chapter 12 Mid-Chapter Test . . . . . . . . . . . . 769Chapter 12 Cumulative Review . . . . . . . . . . 770Chapter 12 Standardized Test Practice . 771–772Unit 4 Test/Review (Ch. 11–12) . . . . . . 773–774

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A39

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 12 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 12 Resource Masters includes the core materialsneeded for Chapter 12. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 12-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 12Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 694–695. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

1212

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 12.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

binomial experiment

combination

compound event

dependent and independent events

inclusive events

ihn·KLOO·sihv

margin of sampling error

measure of central tendency

measure of variation

mutually exclusive events

MYOO·chuh·lee

normal distribution

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

odds

permutation

PUHR·myoo·TAY·shuhn

probability

probability distribution

random variable

relative-frequency histogram

sample space

skewed distribution

SKYOOD

standard deviation

variance

VEHR·ee·uhn(t)s

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionThe Counting Principle

NAME ______________________________________________ DATE ____________ PERIOD _____

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© Glencoe/McGraw-Hill 699 Glencoe Algebra 2

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Independent Events If the outcome of one event does not affect the outcome ofanother event and vice versa, the events are called independent events.

Fundamental If event M can occur in m ways and is followed by event N that can occur in n ways, Counting Principle then the event M followed by the event N can occur in m � n ways.

FOOD For the Breakfast Special at the Country Pantry, customerscan choose their eggs scrambled, fried, or poached, whole wheat or white toast,and either orange, apple, tomato, or grapefruit juice. How many differentBreakfast Specials can a customer order?A customer’s choice of eggs does not affect his or her choice of toast or juice, so the eventsare independent. There are 3 ways to choose eggs, 2 ways to choose toast, and 4 ways tochoose juice. By the Fundamental Counting Principle, there are 3 � 2 � 4 or 24 ways tochoose the Breakfast Special.

Solve each problem.

1. The Palace of Pizza offers small, medium, or large pizzas with 14 different toppingsavailable. How many different one-topping pizzas do they serve? 42

2. The letters A, B, C, and D are used to form four-letter passwords for entering a computerfile. How many passwords are possible if letters can be repeated? 256

3. A restaurant serves 5 main dishes, 3 salads, and 4 desserts. How many different mealscould be ordered if each has a main dish, a salad, and a dessert? 60

4. Marissa brought 8 T-shirts and 6 pairs of shorts to summer camp. How many differentoutfits consisting of a T-shirt and a pair of shorts does she have? 48

5. There are 6 different packages available for school pictures. The studio offers 5 differentbackgrounds and 2 different finishes. How many different options are available? 60

6. How many 5-digit even numbers can be formed using the digits 4, 6, 7, 2, 8 if digits canbe repeated? 2500

7. How many license plate numbers consisting of three letters followed by three numbersare possible when repetition is allowed? 17,576,000

8. How many 4-digit positive even integers are there? 4500

ExampleExample

ExercisesExercises


Dependent Events If the outcome of an event does affect the outcome of another event,the two events are said to be dependent. The Fundamental Counting Principle still applies.

ENTERTAINMENT The guests at a sleepover brought 8 videos. Theydecided they would only watch 3 videos. How many orders of 3 different videosare possible?After the group chooses to watch a video, they will not choose to watch it again, so thechoices of videos are dependent events.

There are 8 choices for the first video. That leaves 7 choices for the second. After they choosethe first 2 videos, there are 6 remaining choices. Thus by the Fundamental CountingPrinciple, there are 8 � 7 � 6 or 336 orders of 3 different videos.

Solve each problem.

1. Three students are scheduled to give oral reports on Monday. In how many ways cantheir presentations be ordered? 6

2. In how many ways can the first five letters of the alphabet be arranged if each letter isused only once? 120

3. In how many different ways can 4 different books be arranged on the shelf? 24

4. How many license plates consisting of three letters followed by three numbers arepossible when no repetition is allowed? 11,232,000

5. Sixteen teams are competing in a soccer match. Gold, silver, and bronze medals will beawarded to the top three finishers. In how many ways can the medals be awarded? 3360

6. In a word-building game each player picks 7 letter tiles. If Julio’s letters are all different,how many 3-letter combinations can he make out of his 7 letters? 210

7. The editor has accepted 6 articles for the news letter. In how many ways can the 6 articlesbe ordered? 720

8. There are 10 one-hour workshops scheduled for the open house at the greenhouse.There is only one conference room available. In how many ways can the workshops beordered? 3,628,800

9. The top 5 runners at the cross-country meet will receive trophies. If there are 22 runnersin the race, in how many ways can the trophies be awarded? 3,160,080

Study Guide and Intervention (continued)

The Counting Principle

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExampleExample

ExercisesExercises

Skills PracticeThe Counting Principle

NAME ______________________________________________ DATE ____________ PERIOD _____

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State whether the events are independent or dependent.

1. finishing in first, second, or third place in a ten-person race dependent

2. choosing a pizza size and a topping for the pizza independent

3. Seventy-five raffle tickets are placed in a jar. Three tickets are then selected, one afterthe other, without replacing a ticket after it is chosen. dependent

4. The 232 members of the freshman class all vote by secret ballot for the classrepresentative to the Student Senate. independent

Solve each problem.

5. A surveying firm plans to buy a color printer for printing its maps. It has narrowed itschoice to one of three models. Each of the models is available with either 32 megabytesof random access memory (RAM), 64 megabytes of RAM, or 128 megabytes of RAM.From how many combinations of models and RAM does the firm have to choose? 9

6. How many arrangements of three letters can be formed from the letters of the wordMATH if any letter will not be used more than once? 24

7. Allan is playing the role of Oliver in his school’s production of Oliver Twist. Thewardrobe crew has presented Allan with 5 pairs of pants and 4 shirts that he can wear.How many possible costumes consisting of a pair of pants and a shirt does Allan have tochoose from? 20

8. The 10-member steering committee that is preparing a study of the public transportationneeds of its town will select a chairperson, vice-chairperson, and secretary from thecommittee. No person can serve in more than one position. In how many ways can thethree positions be filled? 720

9. Jeanine has decided to buy a pickup truck. Her choices include either a V-6 engine or aV-8 engine, a standard cab or an extended cab, and 2-wheel drive or 4-wheel drive. Howmany possible models does she have to choose from? 8

10. A mail-order company that sells gardening tools offers rakes in two different lengths.Customers can also choose either a wooden, plastic, or fiberglass handle for the rake.How many different kinds of rakes can a customer buy? 6

11. A Mexican restaurant offers chicken, beef, or vegetarian fajitas wrapped with either cornor flour tortillas, and topped with either mild, medium, or hot salsa. How many differentchoices of fajitas does a customer have? 18


State whether the events are independent or dependent.

1. choosing an ice cream flavor and choosing a topping for the ice cream independent

2. choosing an offensive player of the game and a defensive player of the game in aprofessional football game independent

3. From 15 entries in an art contest, a camp counselor chooses first, second, and third placewinners. dependent

4. Jillian is selecting two more courses for her block schedule next semester. She mustselect one of three morning history classes and one of two afternoon math classes.independent

Solve each problem.

5. A briefcase lock has 3 rotating cylinders, each containing 10 digits. How many numericalcodes are possible? 1000

6. A golf club manufacturer makes irons with 7 different shaft lengths, 3 different grips, 5different lies, and 2 different club head materials. How many different combinations areoffered? 210

7. There are five different routes that a commuter can take from her home to the office. Inhow many ways can she make a round trip if she uses a different route coming thangoing? 20

8. In how many ways can the four call letters of a radio station be arranged if the firstletter must be W or K and no letters repeat? 27,600

9. How many 7-digit phone numbers can be formed if the first digit cannot be 0 or 1, andany digit can be repeated? 8,000,000

10. How many 7-digit phone numbers can be formed if the first digit cannot be 0, and anydigit can be repeated? 9,000,000

11. How many 7-digit phone numbers can be formed if the first digit cannot be 0 or 1, and ifno digit can be repeated? 483,840

12. How many 7-digit phone numbers can be formed if the first digit cannot be 0, and if nodigit can be repeated? 544,320

13. How many 6-character passwords can be formed if the first character is a digit and theremaining 5 characters are letters that can be repeated? 118,813,760

14. How many 6-character passwords can be formed if the first and last characters aredigits and the remaining characters are letters? Assume that any character can berepeated. 45,697,600

Practice (Average)

The Counting Principle

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

Reading to Learn MathematicsThe Counting Principle

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How can you count the maximum number of license plates a statecan issue?

Read the introduction to Lesson 12-1 at the top of page 632 in your textbook.

Assume that all Florida license plates have three letters followed by threedigits, and that there are no rules against using the same letter or numbermore than once. How many choices are there for each letter? for each digit?26; 10

Reading the Lesson

1. Shamim is signing up for her classes. Most of her classes are required, but she has twoelectives. For her arts class, she can chose between Art, Band, Chorus, or Drama. For herlanguage class, she can choose between French, German, and Spanish.

a. To organize her choices, Shamim decides to make a tree diagram. Let A, B, C, and Drepresent Art, Band, Chorus, and Drama, and F, G, and S represent French, German,and Spanish. Complete the following diagram.

b. How could Shamim have found the number of possible combinations without making atree diagram? Sample answer: Multiply the number of choices for herarts class by the number of choices for her language class: 3 � 4 �12.

2. A jar contains 6 red marbles, 4 blue marbles, and 3 yellow marbles. Indicate whether theevents described are dependent or independent.

a. A marble is drawn out of the jar and is not replaced. A second marble is drawn.dependent

b. A marble is drawn out of the jar and is put back in. The jar is shaken. A secondmarble is drawn. independent

Helping You Remember

3. One definition of independent is “not determined or influenced by someone or somethingelse.” How can this definition help you remember the difference between independentand dependent events? Sample answer: If the outcome of one event does notaffect or influence the outcome of another, the events are independent. Ifthe outcome of one event does affect or influence the outcome ofanother, the events are dependent.


Tree Diagrams and the Power RuleIf you flip a coin once, there are two possible outcomes: heads showing (H) or tails showing (T).The tree diagram to the right shows the four (22)possible outcomes if you flip a coin twice.

Flip 2

HTHT

Flip 1

H

T

Outcomes

HHHTTHTT

start

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

Draw a tree diagram toshow all the possible outcomes for flippinga coin three times. List the outcomes.

There are eight (23) possible outcomes. Witheach extra flip, the number of outcomes doubles. With 4 flips, there would be sixteen(24) outcomes.

Flip 2

H

T

H

T

Flip 1

H

T

Flip 3

HTHTHTHT

Outcomes

HHHHHTHTHHTTTHHTHTTTHTTT

start

In a cup there are ared, a blue, and a yellow marble. Howmany possible outcomes are there ifyou draw one marble at random,replace it, and then draw another?

There are nine (32) possible outcomes.

Draw 2

RBYRBYRBY

Outcomes

RRRBRYBRBBBYYRYBYY

Draw 1

R

B

Y

start

Example 1Example 1 Example 2Example 2

The Power Rule for the number of outcomes states that if an experiment isrepeated n times, and if there are b possible outcomes each time, there are bn total possible outcomes.

Find the total number of possible outcomes for each experiment. Usetree diagrams to help you.

1. flipping a coin 5 times 2. doing the marble experiment 6 times

3. flipping a coin 8 times 4. rolling a 6-sided die 2 times

5. rolling a 6-sided die 3 times 6. rolling a 4-sided die 2 times

7. rolling a 4-sided die 3 times 8. rolling a 12-sided die 2 times

Study Guide and InterventionPermutations and Combinations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Permutations When a group of objects or people are arranged in a certain order, thearrangement is called a permutation.

Permutations The number of permutations of n distinct objects taken r at a time is given by P(n, r ) � .

Permutations with Repetitions

The number of permutations of n objects of which p are alike and q are alike is .

The rule for permutations with repetitions can be extended to any number of objects thatare repeated.

From a list of 20 books, each student must choose 4 books for bookreports. The first report is a traditional book report, the second a poster, the thirda newspaper interview with one of the characters, and the fourth a timeline of theplot. How many different orderings of books can be chosen?Since each book report has a different format, order is important. You must find the numberof permutations of 20 objects taken 4 at a time.

P(n, r) � Permutation formula

P(20, 4) � n � 20, r � 4

� Simplify.

� Divide by common factors.

� 116,280Books for the book reports can be chosen 116,280 ways.

Evaluate each expression.

1. P(6, 3) 120 2. P(8, 5) 6720 3. P(9, 4) 3024 4. P(11, 6) 332,640

How many different ways can the letters of each word be arranged?

5. MOM 3 6. MONDAY 720 7. STEREO 360

8. SCHOOL The high school chorus has been practicing 12 songs, but there is time for only5 of them at the spring concert. How may different orderings of 5 songs are possible?95,040

20 � 19 � 18 � 17 � 16 � 15 � … � 1��16 � 15 � … � 1

20!�16!

20!��(20 � 4)!

n!�(n � r)!

n!�p !q !

n!�(n � r )!

ExampleExample

ExercisesExercises

1 1 1

1 1 1


Combinations An arrangement or selection of objects in which order is not important iscalled a combination.

Combinations The number of combinations of n distinct objects taken r at a time is given by C(n, r ) � .

SCHOOL How many groups of 4 students can be selected from aclass of 20?Since the order of choosing the students is not important, you must find the number ofcombinations of 20 students taken 4 at a time.

C(n, r) � Combination formula

C(20, 4) � n � 20, r � 4

� or 4845

There are 4845 possible ways to choose 4 students.

In how many ways can you choose 1 vowel and 2 consonants from aset of 26 letter tiles? (Assume there are 5 vowels and 21 consonants.)By the Fundamental Counting Principle, you can multiply the number of ways to select onevowel and the number of ways to select 2 consonants. Only the letters chosen matter, notthe order in which they were chosen, so use combinations.

C(5, 1) One of 5 vowels are drawn.C(21, 2) Two of 21 consonants are drawn.

C(5, 1) � C(21, 2) � � Combination formula

� � Subtract.

� 5 � 210 or 1050 Simplify.

There are 1050 combinations of 1 vowel and 2 consonants.


1. C(5, 3) 10 2. C(7, 4) 35 3. C(15, 7) 6435 4. C(10, 5) 252

5. PLAYING CARDS From a standard deck of 52 cards, in how many ways can 5 cards bedrawn? 2,598,960

6. HOCKEY How many hockey teams of 6 players can be formed from 14 players withoutregard to position played? 3003

7. COMMITTEES From a group of 10 men and 12 women, how many committees of 5 menand 6 women can be formed? 232,848

21!�19!2!

5!�4!

21!��(21 � 2)!2!

5!��(5 � 1)!1!

20!�16!4!

20!��(20 � 4)!4!

n!��(n � r)!r!

n !��(n � r )!r !


Permutations and Combinations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticePermutations and Combinations

NAME ______________________________________________ DATE ____________ PERIOD _____

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1. P(6, 3) 120 2. P(8, 2) 56 3. P(2, 1) 2

4. P(3, 2) 6 5. P(10, 4) 5040 6. P(5, 5) 120

7. C(2, 2) 1 8. C(5, 3) 10 9. C(4, 1) 4

10. C(8, 7) 8 11. C(3, 2) 3 12. C(7, 4) 35

Determine whether each situation involves a permutation or a combination. Thenfind the number of possibilities.

13. seating 8 students in 8 seats in the front row of the school auditoriumpermutation; 40,320

14. introducing the 5 starting players on the Woodsville High School basketball team at thebeginning of the next basketball gamepermutation; 120

15. checking out 3 library books from a list of 8 books for a research papercombination; 56

16. choosing 2 movies to rent from 5 moviescombination; 10

17. the first-, second-, and third-place finishers in a race with 10 contestantspermutation; 720

18. electing 4 candidates to a municipal planning board from a field of 7 candidatescombination; 35

19. choosing 2 vegetables from a menu that offers 6 vegetable choicescombination; 15

20. an arrangement of the letters in the word rhombuspermutation; 5040

21. selecting 2 of 8 choices of orange juice at a storecombination; 28

22. placing a red rose bush, a yellow rose bush, a white rose bush, and a pink rose bush in arow in a planter permutation; 24

23. selecting 2 of 9 kittens at an animal rescue sheltercombination; 36

24. an arrangement of the letters in the word isoscelespermutation; 30,240



1. P(8, 6) 20,160 2. P(9, 7) 181,440 3. P(3, 3) 6

4. P(4, 3) 24 5. P(4, 1) 4 6. P(7, 2) 42

7. C(8, 2) 28 8. C(11, 3) 165 9. C(20, 18) 190

10. C(9, 9) 1 11. C(3, 1) 3 12. C(9, 3) � C(6, 2) 1260

Determine whether each situation involves a permutation or a combination. Thenfind the number of possibilities.

13. selecting a 4-person bobsled team from a group of 9 athletescombination; 126

14. an arrangement of the letters in the word Canadapermutation; 120

15. arranging 4 charms on a bracelet that has a clasp, a front, and a backpermutation; 24

16. selecting 3 desserts from 10 desserts that are displayed on a dessert cart in a restaurantcombination; 120

17. an arrangement of the letters in the word annuallypermutation; 5040

18. forming a 2-person sales team from a group of 12 salespeoplecombination; 66

19. making 5-sided polygons by choosing any 5 of 11 points located on a circle to be the verticescombination; 462

20. seating 5 men and 5 women alternately in a row, beginning with a womanpermutation; 14,400

21. STUDENT GROUPS Farmington High is planning its academic festival. All mathclasses will send 2 representatives to compete in the math bowl. How many differentgroups of students can be chosen from a class of 16 students? 120

22. PHOTOGRAPHY A photographer is taking pictures of a bride and groom and their 6attendants. If she takes photographs of 3 people in a group, how many different groupscan she photograph? 56

23. AIRLINES An airline is hiring 5 flight attendants. If 8 people apply for the job, howmany different groups of 5 attendants can the airline hire? 56

24. SUBSCRIPTIONS A school librarian would like to buy subscriptions to 7 newmagazines. Her budget, however, will allow her to buy only 4 new subscriptions. Howmany different groups of 4 magazines can she choose from the 7 magazines? 35

Practice (Average)

Permutations and Combinations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsPermutations and Combinations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How do permutations and combinations apply to softball?


Suppose that 20 students enter a math contest. In how many ways canfirst, second, and third places be awarded? (Write your answer as a product.Do not calculate the product.) 20 � 19 � 18

Reading the Lesson

1. Indicate whether each situation involves a permutation or a combination.

a. choosing five students from a class to work on a special project combination

b. arranging five pictures in a row on a wall permutation

c. drawing a hand of 13 cards from a 52-card deck combination

d. arranging the letters of the word algebra permutation

2. Write an expression that can be used to calculate each of the following.

a. number of combinations of n distinct objects taken r at a time �(n �

n!r)!r!�

b. number of permutations of n objects of which p are alike and q are alike �pn!q!!

�

c. number of permutations of n distinct objects taken r at a time �(n �

n!r)!

�

3. Five cards are drawn from a standard deck of cards. Suppose you are asked to determinehow many possible hands consist of one heart, two diamonds, and two spades.

a. Which of the following would you use to solve this problem: Fundamental CountingPrinciple, permutations, or combinations? (More than one of these may apply.)

Fundamental Counting Principle, combinations

b. Write an expression that involves the notation P(n, r) and/or C(n, r) that you would useto solve this problem. (Do not do any calculations.)

C(13, 1) � C(13, 2) � C(13, 2)


4. Many students have trouble knowing when to use permutations and when to usecombinations to solve counting problems. How can the idea of order help you toremember the difference between permutations and combinations?

Sample answer: A permutation is an arrangement of objects in whichorder is important. A combination is a selection of objects in which orderis not important.


Combinations and Pascal’s TrianglePascal’s triangle is a special array of numbers invented by Blaise Pascal(1623–1662). The values in Pascal’s triangle can be found using thecombinations shown below.

1. Evaluate the expression in each cell of the triangle.

2. The pattern shows the relationship between C(n, r) and Pascal’s triangle. Ingeneral, it is true that C(n, r) � C(n, r � 1) � C(n � 1, r � 1). Completethe proof of this property. In each step, the denominator has been given.

C(n, r) � C(n, r � 1) � �

� �

� �

�

�

�

�

� C(n � 1, r � 1)

(r � 1)![(n � 1) � (r � 1)]!

(r � 1)!(n � r)!

(r � 1)!(n � r)!

(r � 1)!(n � r)!

(r � 1)!(n � r)!(r � 1)!(n � r)!

(r � 1)!(n � r � 1)!(n � r)r!(n � r)!(r � 1)

(r � 1)!(n � r � 1)!r!(n � r)!

C(1,0) C(1,1)

C(2,0) C(2,1) C(2,2)

C(3,0) C(3,1) C(3,2) C(3,3)

C(4,0) C(4,1) C(4,2) C(4,3) C(4,4)

C(5,0) C(5,1) C(5,2) C(5,3) C(5,4) C(5,5)

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Probability and Odds In probability, a desired outcome is called a success; any otheroutcome is called a failure.

Probability of If an event can succeed in s ways and fail in f ways, then the probabilities of success, P (S ),

Success and and of failure, P (F), are as follows.

Failure P (S ) � and P (F) � .

DefinitionIf an event can succeed in s ways and fail in f ways, then the odds of success and of failure are

of Oddsas follows.Odds of success � s :f Odds of failure � f :s

When 3 coins are tossed, what is the probability that at least 2 are heads?

You can use a tree diagram to find the sample space.Of the 8 possible outcomes, 4 have at least 2 heads. So the

probability of tossing at least 2 heads is �48� or �

12�.

What is the probability of picking 4 fiction books and 2 biographiesfrom a best-seller list that consists of 12 fiction books and 6 biographies?By the Fundamental Counting Principle, the number of successes is C(12, 4) � C(6, 2).The total number of selections, s � f, of 6 books is C(18, 6).

P(4 fiction, 2 biography) � or about 0.40

The probability of selecting 4 fiction books and 2 biographies is about 40%.

Find the odds of an event occurring, given the probability of the event.

1. �37� 3:4 2. �

45� 4:1 3. �1

23� 2:11 4. 1:14

Find the probability of an event occurring, given the odds of the event.

5. 10:1 �1101� 6. 2:5 �

27

� 7. 4:9 �143� 8. 8:3 �

181�

One bag of candy contains 15 red candies, 10 yellow candies, and 6 green candies.Find the probability of each selection.

9. picking a red candy �1351� 10. not picking a yellow candy �

2311�

11. picking a green candy �361� 12. not picking a red candy �

1361�

1�15

C(12, 4) � C(6, 2)��C(18, 6)

HHHHHTHTHHTTTHHTHTTTHTTT

HTHTHTHT

H

T

H

T

H

T

FirstCoin

SecondCoin

ThirdCoin

PossibleOutcomes

f�s � f

s�s � f

Example 1Example 1

Example 2Example 2

ExercisesExercises


Probability Distributions A random variable is a variable whose value is thenumerical outcome of a random event. A probability distribution for a particular randomvariable is a function that maps the sample space to the probabilities of the outcomes in thesample space.

Suppose two dice are rolled. The table and the relative-frequencyhistogram show the distribution of the absolute value of the difference of thenumbers rolled. Use the graph to determine which outcome is the most likely.What is its probability?

The greatest probability in the graph is �14�.

The most likely outcome is a difference of 1 and its

probability is �14�.

Four coins are tossed.

1. Complete the table below to show the probability distribution of the number of heads.

2. Make relative-frequency distribution of the data.

10Heads

Heads in Coin Toss

2 3 4

14

Pro

bab

ility

38

18

116

316

516

Number of Heads 0 1 2 3 4

Probability �116� �

14

� �38

� �14

� �116�

14

00

Pro

bab

ility

Difference

Numbers Showing on the Dice

1 2 3 4 5

16

112

Difference 0 1 2 3 4 5

Probability �16

� �14

� �16

� �16

� �16

� �112�


Probability

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Ahmed is posting 2 photographs on his website. He has narrowed his choices to 4landscape photographs and 3 portraits. If he chooses the two photographs atrandom, find the probability of each selection.

1. P(2 portrait) �17

� 2. P(2 landscape) �27

� 3. P(1 of each) �47

�

The Carubas have a collection of 28 video movies, including 12 westerns and 16 science fiction. Elise selects 3 of the movies at random to bring to a sleep-overat her friend’s house. Find the probability of each selection.

4. P(3 westerns) �85159

� 5. P(3 science fiction) �12107

�

6. P(1 western and 2 science fiction) �4901� 7. P(2 westerns and 1 science fiction) �

28783

�

8. P(3 comedy) 0 9. P(2 science fiction and 2 westerns) 0

For Exercises 10–13, use the chart that shows the class and gender statistics for the students taking an Algebra 1 or Algebra 2 class at La Mesa High School.If a student taking Algebra 1 or Algebra 2 is selected at random, find each probability. Express as decimals rounded to the nearest thousandth.

10. P(sophomore/female) 0.208

11. P(junior/male) 0.143

12. P(freshman/male) 0.136

13. P(freshman/female) 0.145


14. �58� 5:3 15. �

27� 2:5 16. �

35� 3:2

17. �110� 1:9 18. �

56� 5:1 19. �1

52� 5:7


20. 2:1 �23

� 21. 8:9 �187� 22. 4:1 �

45

�

23. 1:9 �110� 24. 2:7 �

29

� 25. 5:9 �154�

Class/Gender Number

Freshman/Male 95

Freshman/Female 101

Sophomore/Male 154

Sophomore/Female 145

Junior/Male 100

Junior/Female 102


A bag contains 1 green, 4 red, and 5 yellow balls. Two balls are selected atrandom. Find the probability of each selection.

1. P(2 red) �125� 2. P(1 red and 1 yellow) �

49

� 3. P(1 green and 1 yellow) �19

�

4. P(2 green) 0 5. P(2 red and 1 yellow) 0 6. P(1 red and 1 green) �445�

A bank contains 3 pennies, 8 nickels, 4 dimes, and 10 quarters. Two coins areselected at random. Find the probability of each selection.

7. P(2 pennies) �1100� 8. P(2 dimes) �

510� 9. P(1 nickel and 1 dime) �

785�

10. P(1 quarter and 1 penny) 11. P(1 quarter and 1 nickel) 12. P(2 dimes and 1 quarter)

�110� �

145� 0

Henrico visits a home decorating store to choose wallpapers for his new house. Thestore has 28 books of wallpaper samples, including 10 books of WallPride samplesand 18 books of Deluxe Wall Coverings samples. The store will allow Henrico tobring 4 books home for a few days so he can decide which wallpapers he wants tobuy. If Henrico randomly chooses 4 books to bring home, find the probability ofeach selection.

13. P(4 WallPride) �1295� 14. P(2 WallPride and 2 Deluxe) �

145535

�

15. P(1 WallPride and 3 Deluxe) �1534645

� 16. P(3 WallPride and 1 Deluxe) �44585

�

For Exercises 17–20, use the table that shows the range of verbal SAT scores forfreshmen at a small liberalarts college. If a freshman student is chosen at random, find each probability.Express as decimals rounded to the nearest thousandth.

17. P(400–449) 0.052 18. P(550–559) 0.243 19. P(at least 650) 0.166


20. �141� 4:7 21. �

1123� 12:1 22. �9

59� 5:94 23. �10

100� 1:999

24. �156� 5:11 25. �9

35� 3:92 26. �7

90� 9:61 27. �1

85� 8:7


28. 2:23 �225� 29. 2:5 �

27

� 30. 15:1 �1156� 31. 9:7 �

196�

32. 11:14 �1215� 33. 1000:1 �

11000001

� 34. 12:17 �1229� 35. 8:13 �

281�

Range 400–449 450–499 500–549 550–559 600–649 650�

Number of Students

129 275 438 602 620 412

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Pre-Activity What do probability and odds tell you about life’s risks?


What is the probability that a person will not be struck by lightning in agiven year?

�774590,,909090

�

Reading the Lesson

1. Indicate whether each of the following statements is true or false.

a. If an event can never occur, its probability is a negative number. false

b. If an event is certain to happen, its probability is 1. true

c. If an event can succeed in s ways and fail in f ways, then the probability of success

is . false

d. If an event can succeed in s ways and fail in f ways, then the odds against the eventare s :f. false

e. A probability distribution is a function in which the domain is the sample space of anexperiment. true

2. A weather forecast says that the chance of rain tomorrow is 40%.

a. Write the probability that it will rain tomorrow as a fraction in lowest terms. �25

�

b. Write the probability that it will not rain tomorrow as a fraction in lowest terms. �35

�

c. What are the odds in favor of rain? 2:3

d. What are the odds against rain? 3:2

3. Refer to the table in Example 4 on page 646 in your textbook.

a. What other sum has the same probability as a sum of 11? 3

b. What are the odds of rolling a sum of 8? 5:31

c. What are the odds against rolling a sum of 9? 8:1


4. A good way to remember something is to explain it to someone else. Suppose that yourfriend Roberto is having trouble remembering the difference between probability andodds. What would you tell him to help him remember this easily?

Sample answer: Probability gives the ratio of successes to the totalnumber of outcomes, while odds gives the ratio of successes to failures.

s�f


Geometric ProbabilityIf a dart, thrown at random, hits the triangular board shown at the right, what is the chance that it will hit the shaded region? This chance, also called a probability, can be determined by comparing the area of the shaded region to the area of the board. This ratio indicates what fraction of the tosses should hit in the shaded region.

�

� �1224� or �

12�

In general, if S is a subregion of some region R, then the probability,P(S), that a point, chosen at random, belongs to subregion S is given by the following.

P(S) �

Find the probability that a point, chosen at random, belongs to theshaded subregions of the following regions.

1. �12

� 2. �59

� 3. ��4

�

The dart board shown at the right has 5 concentric circles whose centers are also the center of the square board. Each side of the board is 38 cm, and the radii of the circles are 2 cm, 5 cm, 8 cm, 11 cm, and 14 cm. A dart hitting within one of the circular regions scores the number of points indicated on the board, while a hit anywhere else scores 0 points. If a dart, thrown at random, hits the board, find the probability of scoring the indicated number of points.

4. 0 points 5. 1 point 6. 2 points

�361

3�61

49��

17454�4

� �15474�4

�

7. 3 points 8. 4 points 9. 5 points

�13494�4

� �12414�4

� �3

�61�

51

2

34

4 4

4

4

46

6

64

4

3 3

5

5

area of subregion S��are of region R

�12

�(4)(6)��12

�(8)(6)

area of shaded region��area of triangular board

4 4

6

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Probability of Independent Events

Probability of Two If two events, A and B, are independent, then the probability of both occurring isIndependent Events P(A and B) � P(A) � P(B).

In a board game each player has 3 different-colored markers. To move around the board the player first spins a spinner to determine which piece can be moved. He or she then rolls a die to determine how many spacesthat colored piece should move. On a given turn what is theprobability that a player will be able to move the yellow piece more than 2 spaces?Let A be the event that the spinner lands on yellow, and let B be the event that the die

shows a number greater than 2. The probability of A is �13�, and the probability of B is �

23�.

P(A and B) � P(A) � P(B) Probability of independent events

� �13� � �

23� or �

29� Substitute and multiply.

The probability that the player can move the yellow piece more than 2 spaces is �29�.

A die is rolled 3 times. Find the probability of each event.

1. a 1 is rolled, then a 2, then a 3 �2116�

2. a 1 or a 2 is rolled, then a 3, then a 5 or a 6 �514�

3. 2 odd numbers are rolled, then a 6 �214�

4. a number less than 3 is rolled, then a 3, then a number greater than 3 �316�

5. A box contains 5 triangles, 6 circles, and 4 squares. If a figure is removed, replaced, anda second figure is picked, what is the probability that a triangle and then a circle will be picked? �

125� or about 0.13

6. A bag contains 5 red marbles and 4 white marbles. A marble is selected from the bag,then replaced, and a second selection is made. What is the probability of selecting 2 redmarbles? �

2851� or about 0.31

7. A jar contains 7 lemon jawbreakers, 3 cherry jawbreakers, and 8 rainbow jawbreakers.What is the probability of selecting 2 lemon jawbreakers in succession providing thejawbreaker drawn first is then replaced before the second is drawn?

�34294

� or about 0.15

blue

redyellow

ExampleExample

ExercisesExercises


Probability of Dependent Events

Probability of Two If two events, A and B, are dependent, then the probability of both events occurring isDependent Events P (A and B ) � P (A) � P (B following A).

There are 7 dimes and 9 pennies in a wallet. Suppose two coins areto be selected at random, without replacing the first one. Find the probability ofpicking a penny and then a dime.Because the coin is not replaced, the events are dependent.

Thus, P(A and B) � P(A) � P(B following A).P(penny, then dime) � P(penny) � P(dime following penny)

�196� � �1

75� � �

2810�

The probability is �2810� or about 0.26

What is the probability of drawing, without replacement, 3 hearts,then a spade from a standard deck of cards?Since the cards are not replaced, the events are dependent. Let H represent a heart and Srepresent a spade.

P(H, H, H, S) � P(H) � P(H following H) � P(H following 2 Hs) � P(S following 3 Hs)

� �1532� � �

1521� � �

1510� � �

1439� or about 0.003

The probability is about 0.003 of drawing 3 hearts, then a spade.

Find each probability.

1. The cup on Sophie’s desk holds 4 red pens and 7 black pens. What is the probability ofher selecting first a black pen, then a red one? �

1545� or about 0.25

2. What is the probability of drawing two cards showing odd numbers from a set of cardsthat show the first 20 counting numbers if the first card is not replaced before thesecond is chosen? �

398� or about 0.24

3. There are 3 quarters, 4 dimes, and 7 nickels in a change purse. Suppose 3 coins areselected without replacement. What is the probability of selecting a quarter, then a dime,and then a nickel? �

216� or about 0.04

4. A basket contains 4 plums, 6 peaches, and 5 oranges. What is the probability of picking 2 oranges, then a peach if 3 pieces of fruit are selected at random? �

941� or about 0.04

5. A photographer has taken 8 black and white photographs and 10 color photographs for abrochure. If 4 photographs are selected at random, what is the probability of picking first2 black and white photographs, then 2 color photographs? �

1702� or about 0.07


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A die is rolled twice. Find each probability.

1. P(5, then 6) �316� 2. P(no 2s) �

2356� 3. P(two 1s) �

316�

4. P(any number, then not 5) �56

� 5. P(4, then not 6) �356� 6. P(not 1, then not 2) �

2356�

A board game uses a set of 6 different cards. Each card displays one of the followingfigures: a star, a square, a circle, a diamond, a rectangle, or a pentagon. The cardsare placed face down, and a player chooses two cards. Find each probability.

7. P(circle, then star), if no replacement occurs �310�

8. P(diamond, then square), if replacement occurs �316�

9. P(2 polygons), if replacement occurs �2356�

10. P(2 polygons), if no replacement occurs �23

�

11. P(circle, then hexagon), if no replacement occurs 0

Determine whether the events are independent or dependent. Then find eachprobability.

12. A mixed box of herbal teabags contains 2 lemon teabags, 3 orange-mango teabags,3 chamomile teabags, and 1 apricot-ginger teabag. Kevin chooses 2 teabags at random tobring to work with him. What is the probability that he first chooses a lemon teabag andthen a chamomile teabag? dependent; �

112�

13. The chart shows the selection of olive oils that Hasha finds in a specialty foods catalog. If sherandomly selects one type of oil, then randomlyselects another, different oil, what is the probability that both selections are domestic,first cold pressed oils? dependent; �8

2210

�

For Exercises 14 and 15, two thirds of the area of the spinner earns you 50 points. Suppose you spin the spinner twice.

14. Sketch a tree diagram showing all of the possibilities. Use it to find the probability ofspinning 50 points, then 100 points. �

29

�

15. What is the probability that you get 100 points on each spin? �

19

�

100

50

Type of Oil Domestic Imported

Pure 2 5

Cold Pressed 4 8

First Cold Pressed 7 15


A die is rolled three times. Find each probability.

1. P(three 4s) �2116� 2. P(no 4s) �

122156

�

3. P(2, then 3, then 1) �2116� 4. P(three different even numbers) �

316�

5. P(any number, then 5, then 5) �316� 6. P(even number, then odd number, then 1) �

214�

There are 3 nickels, 2 dimes, and 5 quarters in a purse. Three coins are selected insuccession at random. Find the probability.

7. P(nickel, then dime, then quarter), if no replacement occurs �214�

8. P(nickel, then dime, then quarter), if replacement occurs �1300�

9. P(2 nickels, then 1 quarter), if no replacement occurs �214�

10. P(3 dimes), if replacement occurs �1125�

11. P(3 dimes), if no replacement occurs 0

For Exercises 12 and 13, determine whether the events are independent ordependent. Then find each probability.

12. Serena is creating a painting. She wants to use 2 more colors. She chooses randomly from6 shades of red, 10 shades of green, 4 shades of yellow, 4 shades of purple, and 6 shadesof blue. What is the probability that she chooses 2 shades of green? dependent; �

239�

13. Kershel’s mother is shopping at a bakery. The owner offers Kershel a cookie from a jarcontaining 22 chocolate chip cookies, 18 sugar cookies, and 15 oatmeal cookies. Withoutlooking, Kershel selects one, drops it back in, and then randomly selects another. What isthe probability that neither selection was a chocolate chip cookie? independent; �

295�

14. METEOROLOGY The Fadeeva’s are planning a 3-day vacation to the mountains. Along-range forecast reports that the probability of rain each day is 10%. Assuming thatthe daily probabilities of rain are independent, what is the probability that there is norain on the first two days, but that it rains on the third day? �

180100�

RANDOM NUMBERS For Exercises 15 and 16, use the following information.Anita has a list of 20 jobs around the house to do, and plans to do 3 of them today. She assigns each job a number from 1 to 20, andsets her calculator to generate random numbers from 1 to 20, whichcan reoccur. Of the jobs, 3 are outside, and the rest are inside.

15. Sketch a tree diagram showing all of the possibilities that the first three numbers generated correspond to inside jobs or outside jobs. Use it to find the probability that the first two numbers correspond to inside jobs,and the third to an outside job. 0.108375

16. What is the probability that the number generated corresponds to an outside job three times in a row? 0.003375

Practice (Average)

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Pre-Activity How does probability apply to basketball?


Write the probability that Reggie Miller made a free-throw shot during the1998�99 season as a fraction in lowest terms. (Your answer should notinclude a decimal.) �

128030

�

Reading the Lesson

1. A bag contains 4 yellow balls, 5 red balls, 1 white ball, and 2 black balls. A ball is drawnfrom the bag and is not replaced. A second ball is drawn.

a. Let Y be the event “first ball is yellow” and B be the event “second ball is black.” Arethese events independent or dependent? dependent

b. Tell which formula you would use to find the probability that the first ball is yellowand the second ball is black. C

A. P(Y and B) �

B. P(Y and B) � P(Y) � P(B)

C. P(Y and B) � P(Y) � P(B following Y)

c. Which equation shows the correct calculation of this probability? B

A. �13� � �1

21� � �

1373� B. �

13� � �1

21� � �3

23�

C. �13� � �

16� � �

12� D. �

13� � �

16� � �1

18�

d. Which equation shows the correct calculation of the probability that if three balls aredrawn in succession without replacement, all three will be red? B

A. �152� � �1

52� � �1

52� � �1

172258� B. �1

52� � �1

41� � �1

30� � �2

12�

C. �152� � �1

41� � �1

30� � �

761630�


2. Some students have trouble remembering a lot of formulas, so they try to keep thenumber of formulas they have to know to a minimum. Can you learn just one formulathat will allow you to find probabilities for both independent and dependent events?Explain your reasoning. Sample answer: Just remember the formula fordependent events: P(A and B) � P(A) � P(B following A). When theevents are independent, P(B following A) � P(B), so the formula fordependent events simplifies to P(A and B) � P(A) � P(B), which is thecorrect formula for independent events.

P(Y)��P(Y) � P(B)


Conditional ProbabilitySuppose a pair of dice is thrown. It is known that the sum is greater thanseven. Find the probability that the dice match.

The probability of an event given the occurrence of another event is calledconditional probability. The conditional probability of event A, the dicematch, given event B, their sum is greater than seven, is denoted P(A/B).

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There are 15 sums greater than seven andthere are 36 possible pairs altogether.

P(B) � �1356�

There are three matching pairs greaterthan seven.

P(A and B) � �336�

P(A/B) �

P(A/B) � or �15�

The conditional probability is �15�.

A card is drawn from a standard deck of 52 and is found to be red.Given that event, find each of the following probabilities.

1. P(heart) 2. P(ace) 3. P(face card)

4. P(jack or ten) 5. P(six of spades) 6. P(six of hearts)

A sports survey taken at Stirers High School shows that 48% of therespondents liked soccer, 66% liked basketball, and 38% liked hockey.Also, 30% liked soccer and basketball, 22% liked basketball and hockeyand 28% liked soccer and hockey. Finally, 12% liked all three sports.Find each of the following probabilities.

7. The probability Meg likes soccer if she likes basketball.

8. The probability Biff likes basketball if he likes soccer.

9. The probability Muffy likes hockey if she likes basketball.

10. The probability Greg likes hockey and basketball if he likes soccer.

�336�

�

�1356�

P(A and B)��P(B)

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Mutually Exclusive Events Events that cannot occur at the same time are calledmutually exclusive events.

Probability of Mutually If two events, A and B, are mutually exclusive, thenExclusive Events P(A or B ) � P(A) � P(B ).

This formula can be extended to any number of mutually exclusive events.

To choose an afternoon activity, summer campers pull slips ofpaper out of a hat. Today there are 25 slips for a nature walk, 35 slips forswimming, and 30 slips for arts and crafts. What is the probability that a camperwill pull a slip for a nature walk or for swimming?These are mutually exclusive events. Note that there is a total of 90 slips.

P(nature walk or swimming) � P(nature walk) � P(swimming)

� �2950� � �

3950� or �

23�

The probability of a camper’s pulling out a slip for a nature walk or for swimming is �23�.

By the time one tent of 6 campers gets to the front of the line, thereare only 10 nature walk slips and 15 swimming slips left. What is the probabilitythat more than 4 of the 6 campers will choose a swimming slip?

P(more than 4 swimmers) � P(5 swimmers) � P(6 swimmers)

� �

� 0.2The probability of more than 4 of the campers swimming is about 0.2.


1. A bag contains 45 dyed eggs: 15 yellow, 12 green, and 18 red. What is the probability ofselecting a green or a red egg? �

23

�

2. The letters from the words LOVE and LIVE are placed on cards and put in a box. Whatis the probability of selecting an L or an O from the box? �

38

�

3. A pair of dice is rolled, and the two numbers are added. What is the probability that thesum is either a 5 or a 7? �

158� or about 0.28

4. A bowl has 10 whole wheat crackers, 16 sesame crackers, and 14 rye crisps. If a personpicks a cracker at random, what is the probability of picking either a sesame cracker ora rye crisp? �

34

�

5. An art box contains 12 colored pencils and 20 pastels. If 5 drawing implements are chosenat random, what is the probability that at least 4 of them are pastels? about 0.37

C(10, 0) � C(15, 6)��C(25, 6)

C(10, 1) � C(15, 5)��C(25, 6)

Example 1Example 1

Example 2Example 2

ExercisesExercises


Inclusive Events

Probability of Inclusive Events If two events, A and B, are inclusive, P(A or B ) � P(A) � P(B ) � P(A and B ).

What is the probability of drawing a face card or a black card froma standard deck of cards?The two events are inclusive, since a card can be both a face card and a black card.

P(face card or black card) � P(face card) � P(black card) � P(black face card)

� �133� � �

12� � �2

36�

� �183� or about 0.62

The probability of drawing either a face card or a black card is about 0.62


1. What is the probability of drawing a red card or an ace from a standard deck of cards?

�173� or about 0.54

2. Three cards are selected from a standard deck of 52 cards. What is the probability ofselecting a king, a queen, or a red card?

�1256� or about 0.58

3. The letters of the alphabet are placed in a bag. What is the probability of selecting avowel or one of the letters from the word QUIZ?

�276� or about 0.27

4. A pair of dice is rolled. What is the probability that the sum is odd or a multiple of 3?

�171� or about 0.64

5. The Venn diagram at the right shows the number of juniors on varsity sports teams at Elmwood High School.Some athletes are on varsity teams for one season only,some athletes for two seasons, and some for all threeseasons. If a varsity athlete is chosen at random from the junior class, what is the probability that he or she plays a fall or winter sport? �

1136�

Winter

Juniors Playing Varsity Sports

Spring

Fall5

6

8 3

54 1


Adding Probabilities

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Eli has 10 baseball cards of 10 different players in his pocket. Three players arepitchers, 5 are outfielders, and 2 are catchers. If Eli randomly selects a card totrade, find each probability.

1. P(pitcher or outfielder) �45

� 2. P(pitcher or catcher) �12

� 3. P(outfielder or catcher) �170�

A die is rolled. Find each probability.

4. P(5 or 6) �13

� 5. P(at least a 3) �23

� 6. P(less than 4) �12

�

Determine whether the events are mutually exclusive or inclusive. Then find theprobability.

7. A die is rolled. What is the probability of rolling a 3 or a 4? mutually exclusive; �13

�

8. A die is rolled. What is the probability of rolling an even number or a 4? inclusive; �12

�

9. A card is drawn from a standard deck of cards. What is the probability of drawing a kingor a queen? mutually exclusive; �

123�

10. A card is drawn from a standard deck of cards. What is the probability of drawing a jackor a heart? inclusive; �

143�

11. The sophomore class is selling Mother’s Day plants to raise money. Susan’s prize forbeing the top seller of plants is a choice of a book, a CD, or a video. She can choose from6 books, 3 CDs, and 5 videos. What is the probability that Susan selects a book or a CD?

mutually exclusive; �194�

A spinner numbered 1�10 is spun. Find each probability.

12. P(less than 5 or even) �170� 13. P(even or odd) 1 14. P(prime or even) �

45

�

Two cards are drawn from a standard deck of cards. Find each probability.

15. P(both red or both black) �2551� 16. P(both aces or both red) �

25251

�

17. P(both 2s or both less than 5) �21211

� 18. P(both black or both less than 5) �168683

�

For Exercises 19 and 20, use the Venn diagram that shows the number of participants in two different kinds of aerobic exercise classes that are offered at a health club. Determine each probability if a person is selected at random from the participants.

19. P(step aerobics or jazzercise, but not both) �4692�

20. P(step aerobics and jazzercise) �1632�

JazzerciseStep

Aerobics

2722

13


An urn contains 7 white marbles and 5 blue marbles. Four marbles are selectedwithout replacement. Find each probability.

1. P(4 white or 4 blue) �989� 2. P(exactly 3 white) �

3959� 3. P(at least 3 white) �

1343�

4. P(fewer than 3 white) �1393� 5. P(3 white or 3 blue) �

4999� 6. P(no white or no blue) �

989�

Jason and Maria are playing a board game in which three dice are tossed todetermine a player’s move. Find each probability.

7. P(two 5s) �752� 8. P(three 5s) �

2116� 9. P(at least two 5s) �

227�

10. P(no 5s) �122156

� 11. P(one 5) �2752� 12. P(one 5 or two 5s) �

152�

Determine whether the events are mutually exclusive or inclusive. Then find theprobability.

13. A clerk chooses 4 CD players at random for floor displays from a shipment of 24 CD players.If 15 of the players have a blue case and the rest have a red case, what is the probability ofchoosing 4 players with a blue case or 4 players with a red case? mutual. exclus.; �

57016

�

14. A department store employs 28 high school students, all juniors and seniors. Six of the12 seniors are females and 12 of the juniors are males. One student employee is chosenat random. What is the probability of selecting a senior or a female? inclusive; �

47

�

15. A restaurant has 5 pieces of apple pie, 4 pieces of chocolate cream pie, and 3 pieces ofblueberry pie. If Janine selects a piece of pie at random for dessert, what is theprobability that she selects either apple or chocolate cream? mutually exclusive; �

34

�

16. At a statewide meeting, there are 20 school superintendents, 13 principals, and 6 assistantprincipals. If one of these people is chosen at random, what is the probability that he orshe is either a principal or an assistant principal? mutually exclusive; �

1399�

17. An airline has one bank of 13 telephones at a reservations office. Of the 13 operators whowork there, 8 take reservations for domestic flights and 5 take reservations for internationalflights. Seven of the operators taking domestic reservations and 3 of the operators takinginternational reservations are female. If an operator is chosen at random, what is theprobability that the person chosen takes domestic reservations or is a male?

inclusive; �1103�

18. MUSIC Forty senior citizens were surveyed about their music preferences. The results are displayed in the Venndiagram. If a senior citizen from the survey group isselected at random, what is the probability that he or she likes only country and western music? What is theprobability that he or she likes classical and/or country,but not 1940’s pop?�230�; �

25

�

Countryand

Western

1940’s Pop

Classical

6

9

3 7

65 4

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Pre-Activity How does probability apply to your personal habits?


Why do the percentages shown on the bar graph add up to more than100%? Sample answer: Many people do more than one of thelisted bedtime rituals.

Reading the Lesson

1. Indicate whether the events in each pair are inclusive or mutually exclusive.

a. Q: drawing a queen from a standard deck of cardsD: drawing a diamond from a standard deck of cards inclusive

b. J: drawing a jack from a standard deck of cardsK: drawing a king from a standard deck of cards mutually exclusive

2. Marla took a quiz on this lesson that contained the following problem.Each of the integers from 1 through 25 is written on a slip of paper and placed in anenvelope. If one slip is drawn at random, what is the probability that it is odd or amultiple of 5?Here is Marla’s work.

P(odd) � �1235� P(multiple of 5) � �2

55� or �

15�

P(odd or multiple of 5) � P(odd) � P(multiple of 5)

� �1235� � �2

55� � �

1285�

a. Why is Marla’s work incorrect? Sample answer: Marla used the formula formutually exclusive events, but the events are inclusive. She shoulduse the formula for inclusive events so that the odd multiples of 5 willnot be counted twice.

b. Show the corrected work.

P(odd or multiple of 5) � P(odd) � P(multiple of 5) � P(odd multiple of 5)

� �1235� � �

255� � �

235� � �

1255� � �

35

�


3. Some students have trouble remembering a lot of formulas, so they try to keep thenumber of formulas they have to know to a minimum. Can you learn just one formulathat will allow you to find probabilities for both mutually exclusive and inclusive events?Explain your reasoning. Sample answer: Just remember the formula forinclusive events: P(A or B) � P(A) � P(B) � P(A and B). When theevents are mutually exclusive, P(A and B) � 0, so the formula forinclusive events simplifies to P(A and B) � P(A) � P(B), which is thecorrect formula for mutually exclusive events.


Probability and Tic-Tac-ToeWhat would be the chances of winning at tic-tac-toe if it were turned into agame of pure chance? To find out, the nine cells of the tic-tac-toe board arenumbered from 1 to 9 and nine chips (also numbered from 1 to 9) are putinto a bag. Player A draws a chip at random and enters an X in thecorresponding cell. Player B does the same and enters an O.

To solve the problem, assume that both players draw all their chips withoutlooking and all X and O entries are made at the same time. There are fourpossible outcomes: a draw, A wins, B wins, and either A or B can win.

There are 16 arrangements that result in a draw. Reflections and rotationsmust be counted as shown below.

o x o x o x o o xx o x 4 o o x 4 x x o 8x o x x x o o x x

There are 36 arrangements in which either player may win because bothplayers have winning triples.

x x x x x x x o x x x x x x x x x oo o o 4 x o x 4 x x x 4 x x o 8 o o o 8 x x x 8x o x o o o o o o o o o x x o o o o

In these 36 cases, A’s chances of winning are �1430�.

1. Find the 12 arrangements in which B wins and A cannot.

2. Below are 12 of the arrangements in which A wins and B cannot. Writethe numbers to show the reflections and rotations for each arrangement.What is the total number?

o x o x o x x x x x x x x o o x o ox x x o x o x o o o x o x x x x x oo x o x o x x o o o x o o o x o o x

x x o x x x x x x x x x x o o x x oo x x o x o x o o x o o x x x o x oo o x o o x o x o o o x o x o x o x

3. There are �(59!4!!)� different and equally probable

distributions. Complete the chart to find the probability for a draw or for A or B to win.

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Draw: �

A wins: � �1430��

B wins: � �

36�126

16�126

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6Measures of Central Tendency

Use When

Measures of mean the data are spread out and you want an average of values

Central Tendency median the data contain outliers

mode the data are tightly clustered around one or two values

Find the mean, median, and mode of the following set of data:{42, 39, 35, 40, 38, 35, 45}.To find the mean, add the values and divide by the number of values.

mean � � 39.14.

To find the median, arrange the values in ascending or descending order and choose themiddle value. (If there is an even number of values, find the mean of the two middle values.)In this case, the median is 39.To find the mode, take the most common value. In this case, the mode is 35.

Find the mean, median, and mode of each set of data. Round to the nearesthundredth, if necessary.

1. {238, 261, 245, 249, 255, 262, 241, 245} 249.5; 247; 245

2. {9, 13, 8, 10, 11, 9, 12, 16, 10, 9} 10.7; 10; 9

3. {120, 108, 145, 129, 102, 132, 134, 118, 108, 142} 123.8; 124.5; 108

4. {68, 54, 73, 58, 63, 72, 65, 70, 61} 64.89; 65; no mode

5. {34, 49, 42, 38, 40, 45, 34, 28, 43, 30} 38.3; 39; 34

6. The table at the right shows the populations of the six New England capitals. Which would be themost appropriate measure of central tendency to represent the data? Explain why and find that value.Source: www.factfinder.census.gov There is no mode. Thepopulation of Boston is an outlier and would raise the mean too high. The median,79,500, would be the best choice.

CityPopulation (roundedto the nearest 1000)

Augusta, ME 19,000

Boston, MA 589,000

Concord, NH 37,000

Hartford, CT 122,000

Montpelier, VT 8,000

Providence, RI 174,000

42 � 39 � 35 � 40 � 38 � 35 � 45��7

ExampleExample

ExercisesExercises


Measures of Variation The range and the standard deviation measure howscattered a set of data is.

Standard If a set of data consists of the n values x1, x2, …, xn and has mean x�, then the standard deviation

Deviationis given by � � ��.

The square of the standard deviation is called the variance.

Find the variance and standard deviation of the data set {10, 9, 6, 9, 18, 4, 8, 20}.Step 1 Find the mean.

x� � � 10.5

Step 2 Find the variance.

�2 � Standard variance formula

�

� or 27.5

Step 3 Find the standard deviation.� � �27.5�

� 5.2

The variance is 27.5 and the standard deviation is about 5.2.

Find the variance and standard deviation of each set of data. Round to thenearest tenth.

1. {100, 89, 112, 104, 96, 108, 93} 2. {62, 54, 49, 62, 48, 53, 50}58.5; 7.6 29.4; 5.4

3. {8, 9, 8, 8, 9, 7, 8, 9, 6} 4. {4.2, 5.0, 4.7, 4.5, 5.2, 4.8, 4.6, 5.1}0.9; 0.9 0.1; 0.3

5. The table at the right lists the prices of ten brands of breakfast cereal. What is the standard deviation of the values to the nearest penny? $0.33

Price of 10 Brandsof Breakfast Cereal

$2.29 $3.19

$3.39 $2.79

$2.99 $3.09

$3.19 $2.59

$2.79 $3.29

220�8

(10 � 10.5)2 � (9 � 10.5)2 � … � (20 � 10.5)2��8

(x1 � x�)2 � (x2 � x�)2 � … � (xn � x�)2��n

10 � 9 � 6 � 9 � 18 � 4 � 8 � 20��8

(x1 � x�)2 � (x2 � x�)2 � … � (xn � x�)2

��n


Statistical Measures

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6Find the variance and standard deviation of each set of data to the nearest tenth.

1. {32, 41, 35, 35, 46, 42} 23.6, 4.9

2. {13, 62, 77, 24, 38, 19, 88} 763.8, 27.6

3. {89, 99, 42, 16, 42, 71, 16} 959.1, 31.0

4. {450, 400, 625, 225, 300, 750, 650, 625} 30,537.1; 174.7

5. {17, 23, 65, 94, 33, 33, 33, 8, 57, 75, 44, 12, 11, 68, 39} 630.7, 25.1

6. {7.2, 3.1, 3.8, 9.5, 8.3, 8.4} 5.8, 2.4

7. {1.5, 2.5, 3.5, 4.5, 4.5, 5.5, 6.5, 7.5} 3.5, 1.9

For Exercises 8 and 9, use the table that shows the profit in billions of dollarsreported by U.S. manufacturers for the first quarter of the years from 1997through 2001.

Source: U. S. Census Bureau

8. Find the mean and median of the data to the nearest tenth. $64.3 billion, $61.4 billion

9. Which measure of central tendency best represents the data? Explain.The median is more representative because the value 45.3 is not close tothe other data points, and it lowers the mean.

For Exercises 10 and 11, use the table that shows the percent of fourth gradestudents reading at or above the proficiency level in a nationally-administeredreading assessment.

Source: National Center for Education Statistics

10. Find the mean, median, and standard deviation of the data to the nearest tenth.30.5%, 30.5%, 1.1

11. What do the statistics from Exercise 11 tell you about the data?Sample answer: Since the median and mean are equal and the standarddeviation is small, the percent of students reading at or above theproficiency level has not varied much from 1992 to 2000.

Year 1992 1994 1998 2000

Percent at or above proficiency level

29% 30% 31% 32%

Year 1997 1998 1999 2000 2001

Seasonally-Adjusted Profit ($ billions)

$61.4 $75.6 $60.9 $78.5 $45.3


Find the variance and standard deviation of each set of data to the nearest tenth.

1. {47, 61, 93, 22, 82, 22, 37} 2. {10, 10, 54, 39, 96, 91, 91, 18}673.1, 25.9 1228.6, 35.1

3. {1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5} 4. {1100, 725, 850, 335, 700, 800, 950}1.6, 1.2 49,150.0; 221.7

5. {3.4, 7.1, 8.5, 5.1, 4.7, 6.3, 9.9, 8.4, 3.6} 6. {2.8, 0.5, 1.9, 0.8, 1.9, 1.5, 3.3, 2.6, 0.7, 2.5}4.7, 2.2 0.8, 0.9

7. HEALTH CARE Eight physicians with 15 patients on a hospital floor see these patientsan average of 18 minutes a day. The 22 nurses on the same floor see the patients anaverage of 3 hours a day. As a hospital administrator, would you quote the mean,median, or mode as an indicator of the amount of daily medical attention the patients onthis floor receive? Explain. Either the median or the mode; they are equal andhigher than the mean, which is lowered by the smaller amount of timethe physicians spend with the patients.

For Exercises 8–10, use the frequency table that shows the percent of public schoolteachers in the U. S. in 1999 who used computers or theInternet at school for variousadministrative and teachingactivities.

8. Find the mean, median, and modeof the data. 17.75%, 12%, 8%

9. Suppose you believe teachers usecomputers or the Internet tooinfrequently. Which measure would you quote as the “average?” Source: National Assessment of Educational Progress

Explain. Mode; it is lowest.

10. Suppose you believe teachers use computers or the Internet too often. Which measurewould you quote as the “average?” Explain. Mean; it is highest.

For Exercises 11 and 12, use the frequency table that shows the number of games played by 24 American League baseball players between opening day, 2001 andSeptember 8, 2001.

11. Find the mean, median, mode, and standard deviation of thenumber of games played to the nearest tenth.138.2, 138; 138, 2.0

12. For how many players is the number of games within onestandard deviation of the mean? 14

Source: Major League Baseball

No. of Games Frequency

141 4

140 3

139 4

138 5

137 2

136 3

135 3

Percent Using Activity Computer

or Internet

Create instructional materials 39

Administrative record keeping 34

Communicate with colleagues 23

Gather information for planning lessons 16

Multimedia classroom presentations 8

Access research and best practices for teaching 8

Communicate with parents or students 8

Access model lesson plans 6

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6Pre-Activity What statistics should a teacher tell the class after a test?


There is more than one way to give an “average” score for this test. Threemeasures of central tendency for these scores are 94, 76.5 and 73.9. Can youtell which of these is the mean, the median, and the mode without doing anycalculations? Explain your answer.

Sample answer: Yes. The mode must be one of the scores, soit must be an integer. The median must be either one of thescores or halfway between two of the scores, so it must be aninteger or a decimal ending with .5. Therefore, 94 is the mode,76.5 is the median, and 73.9 is the mean.

Reading the Lesson

1. Match each measure with one of the six descriptions of how to find measures of centraltendency and variation.

a. median vi b. mode i c. range iv

d. variance iii e. mean ii f. standard deviation v

i. Find the most commonly occurring values or values in a set of data.

ii. Add the data and divide by the number of items.

iii. Find the mean of the squares of the differences between each value in the set of dataand the mean.

iv. Find the difference between the largest and smallest values in the set of data.

v. Take the positive square root of the variance.

vi. If there is an odd number of items in a set of data, take the middle one. If there is aneven number of items, add the two middle items and divide by 2.


2. It is usually easier to remember a complicated procedure if you break it down into steps.Write the procedure for finding the standard deviation for a set of data in a series ofbrief, numbered steps.

Sample answer: 1. Find the mean. 2. Find the difference between each value and the mean. 3. Square each difference. 4. Find the mean of the squares. 5. Take the positive square root.


Probabilities in GeneticsGenes are the units which transmit hereditary traits. The possible formswhich a gene may take, dominant and recessive, are called alleles. Aparticular trait is determined by two alleles, one from the female parent andone from the male parent. If an organism has the trait which is dominant, itmay have either two dominant alleles or one dominant and one recessiveallele. If the organism has the trait which is recessive, it must have tworecessive alleles.

Consider a plant in which tall stems, T, are dominant toshort stems, t. What is the probability of obtaining a long-stemmedplant if two long-stemmed plants both with the genetic formula Ttare crossed?

A Punnett square is a chart used to determine the possible combinations of characteristics among offspring.

3 tall-stemmed� 1 short-stemmed

4 total

Thus, the probability is �34�.

In a certain plant, red flowers, R, are dominant to white flowers, r.If a white-flowered plant, rr is crossed with a red-flowered plant, Rr,find the probability of each of the following.

1. white-flowered plant �12

� 2. red-flowered plant �12

�

In a certain plant, tall, T, is dominant to short, t, and green pods, G,are dominant to yellow pods, g. Plants with the genetic formulasTtGg and TTGg are crossed. Find the probability of each of thefollowing.

3. tall plant with green pods �34

� 4. tall plant with yellow pods �14

�

TT Tt

T t

Tt

T

t tt

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Normal and Skewed Distributions A continuous probability distribution isrepresented by a curve.

Types of

Normal Positively Skewed Negatively Skewed

ContinuousDistributions

Determine whether the data below appear to be positively skewed,negatively skewed, or normally distributed.{100, 120, 110, 100, 110, 80, 100, 90, 100, 120, 100, 90, 110, 100, 90, 80, 100, 90}Make a frequency table for the data.

Then use the data to make a histogram.Since the histogram is roughly symmetric, the data appear to be normally distributed.

Determine whether the data in each table appear to be positively skewed,negatively skewed, or normally distributed. Make a histogram of the data.

1. {27, 24, 29, 25, 27, 22, 24, 25, 29, 24, 25, 22, 27, 24, 22, 25, 24, 22}positively skewed

2.

normally distributed

3. negatively skewed

�100 101–120

121–140

141–160

161–180

181–200

200�

12

10

8

6

4

2

Freq

uen

cy

Thousands of Dollars

Housing Price No. of Houses Sold

less than $100,000 0

$100,00�$120,000 1

$121,00�$140,000 3

$141,00�$160,000 7

$161,00�$180,000 8

$181,00�$200,000 6

over $200,000 12

104

8

6

4

2Freq

uen

cy

5 6 7 8 9

Shoe Size 4 5 6 7 8 9 10

No. of Students 1 2 4 8 5 1 2

22

6

4

2Freq

uen

cy

24 25 27 29

Value 80 90 100 110 120

Frequency 2 4 7 3 2

80

6

4

2Freq

uen

cy

90 100 110 120

ExampleExample

ExercisesExercises


Use Normal Distributions

Normal Distribution Normal distributions have these properties.The graph is maximized at the mean.The mean, median, and mode are about equal.About 68% of the values are within one standard deviation of the mean.About 95% of the values are within two standard deviations of the mean.About 99% of the values are within three standard deviations of the mean.

The heights of players in a basketball league are normallydistributed with a mean of 6 feet 1 inch and a standard deviation of 2 inches.

a. What is the probability that a player selected at random will be shorter than 5 feet 9 inches?Draw a normal curve. Label the mean and the mean plus or minus multiples of the standard deviation.The value of 5 feet 9 inches is 2 standard deviations below the mean, so approximately 2.5% of the players will be shorter than 5 feet 9 inches.

b. If there are 240 players in the league, about how many players are taller than 6feet 3 inches?The value of 6 feet 3 inches is one standard deviation above the mean. Approximately16% of the players will be taller than this height.240 � 0.16 � 38About 38 of the players are taller than 6 feet 3 inches.

EGG PRODUCTION The number of eggs laid per year by a particular breed ofchicken is normally distributed with a mean of 225 and a standard deviation of 10 eggs.

1. About what percent of the chickens will lay between 215 and 235 eggs per year? 68%

2. In a flock of 400 chickens, about how many would you expect to lay more than 245 eggsper year? 10 chickens

MANUFACTURING The diameter of bolts produced by a manufacturing plant isnormally distributed with a mean of 18 mm and a standard deviation of 0.2 mm.

3. What percent of bolts coming off of the assembly line have a diameter greater than 18.4 mm? 2.5%

4. What percent have a diameter between 17.8 and 18.2 mm? 68%

5'7" 5'9" 5'11" 6'1" 6'3" 6'5" 6'7"

�3

mean

�2 � � �2 �3


The Normal Distribution

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Determine whether the data in each table appear to be positively skewed,negatively skewed, or normally distributed.

1. 2.

normally distributed negatively skewed

For Exercises 3 and 4, use the frequency table that shows the average number of days patients spent on thesurgical ward of a hospital last year.

3. Make a histogram of the data.

4. Do the data appear to be positivelyskewed, negatively skewed, or normally distributed? Explain.Positively skewed; thehistogram is high at the left and has a tail to the right.

DELIVERY For Exercises 5–7, use the following information.The time it takes a bicycle courier to deliver a parcel to his farthest customer is normallydistributed with a mean of 40 minutes and a standard deviation of 4 minutes.

5. About what percent of the courier’s trips to this customer take between 36 and 44 minutes?68%

6. About what percent of the courier’s trips to this customer take between 40 and 48 minutes?47.5%

7. About what percent of the courier’s trips to this customer take less than 32 minutes? 2.5%

TESTING For Exercises 8–10, use the following information.The average time it takes sophomores to complete a math test is normally distributed witha mean of 63.3 minutes and a standard deviation of 12.3 minutes.

8. About what percent of the sophomores take more than 75.6 minutes to complete the test?16%

9. About what percent of the sophomores take between 51 and 63.3 minutes? 34%

10. About what percent of the sophomores take less than 63.3 minutes to complete the test?50%

0–3 4–7 8–11 12–15 16�

2018161412108642

Freq

uen

cy

Days

Patient Stays

Days Number of Patients

0–3 5

4–7 18

8–11 11

12–15 9

16� 6

Speeches Given Political Candidates

0–5 1

6–11 2

12–17 3

18–23 8

24–29 8

Miles Run Track Team Members

0–4 3

5–9 4

10–14 7

15–19 5

20–23 2


Determine whether the data in each table appear to be positively skewed,negatively skewed, or normally distributed.

1. 2.


negatively skewed

For Exercises 3 and 4, use the frequency table that shows the number of hours worked per week by 100 high school seniors.

3. Make a histogram of the data.

4. Do the data appear to be positivelyskewed, negatively skewed, or normally distributed? Explain.Positively skewed; thehistogram is high at the left and has a tail to the right.

TESTING For Exercises 5–10, use the following information.The scores on a test administered to prospective employees are normally distributed with amean of 100 and a standard deviation of 15.

5. About what percent of the scores are between 70 and 130? 95%

6. About what percent of the scores are between 85 and 130? 81.5%

7. About what percent of the scores are over 115? 16%

8. About what percent of the scores are lower than 85 or higher than 115? 32%

9. If 80 people take the test, how many would you expect to score higher than 130? 2

10. If 75 people take the test, how many would you expect to score lower than 85? 12

11. TEMPERATURE The daily July surface temperature of a lake at a resort has a mean of82� and a standard deviation of 4.2�. If you prefer to swim when the temperature is atleast 77.8�, about what percent of the days does the temperature meet your preference?84%

0–8 9–17 18–25 26�

605040302010Fr

equ

ency

Hours

Weekly Work Hours

Hours Number of Students

0–8 30

9–17 45

18–25 20

26� 5

Average Age of High School Principals

Age in Years Number

31–35 3

36–40 8

41–45 15

46–50 32

51–55 40

56–60 38

60� 4

Time Spent at a Museum Exhibit

Minutes Frequency

0–25 27

26–50 46

51–75 89

75–100 57

100� 24

Practice (Average)

The Normal Distribution

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Pre-Activity How are the heights of professional athletes distributed?


There were 53 players on the team and the mean height was approximately73.6. About what fraction of the players’ heights are between 72 and 75,inclusive? Sample answer: about �

23

�

Reading the Lesson

1. Indicate whether each of the following statements is true or false.

a. In a continuous probability distribution, there is a finite number of possible outcomes. false

b. Every normal distribution can be represented by a bell curve. true

c. A distribution that is represented by a curve that is high at the left and has a tail tothe right is negatively skewed. false

d. A normal distribution is an example of a skewed distribution. false

2. Ms. Rose gave the same quiz to her two geometry classes. She recorded the following scores.

First-period class:

Fifth-period class:

In each class, 30 students took the quiz. The mean score for each class was 6.4. Whichset of scores has the greater standard deviation? (Answer this question without doingany calculations.) Explain your answer.

First period class; sample answer: The scores are more spread out fromthe mean than for the fifth period class.


3. Many students have trouble remembering how to determine if a curve represents adistribution that is positively skewed or negatively skewed. What is an easy way toremember this?

Sample answer: Follow the tail! If the tail is on the right (positivedirection), the distribution is positively skewed. If the tail is on the left(negative direction), the distribution is negatively skewed.

Score 0 1 2 3 4 5 6 7 8 9 10

Frequency 0 0 0 0 3 4 9 7 6 1 0

Score 0 1 2 3 4 5 6 7 8 9 10

Frequency 1 0 1 0 3 4 5 7 4 3 2


Street Networks: Finding All Possible RoutesA section of a city is laid out in square blocks. Going north from the intersection of First Avenue and First Street, the avenues are 1st, 2nd, 3rd, and so on. Going east, the streets are numbered in the same way.

Factorials can be used to find the number, r(e, n), of different routes between two intersections. The formula is shown below.

r(e, n) �

The number of streets going east is e; the number of avenues going north is n.

The following problems examine the possible routes from one location to another. Assume that you never use a route that is unnecessarily long.Assume that e 1 and n 1.

Solve each problem.

1. List all the possible routes from 1st Street and 1st Avenue to 4th Streetand 3rd Avenue. Use ordered pairs to show the routes, with streetnumbers first, and avenue numbers second. For example, each routestarts at (1, 1) and ends at (4, 3).

2. Use the formula to compute the number of routes from (1, 1) to (4, 3).There are 4 streets going east and 3 avenues going north.(3

3. Find the number of routes from 1st Street and 1st Avenue to 7th Streetand 6th Avenue.

�6!5

5!)!

� 462

[(e � 1) � (n � 1)]!��(e � 1)!(n � 1)!

6th Ave

5th Ave

4th Ave

3rd Ave

2nd Ave

1st Ave

1st S

t.

2nd

St.

3rd

St.

4th

St.

5th

St.

6th

St.

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Binomial Expansions For situations with only 2 possible outcomes, you can use theBinomial Theorem to find probabilities. The coefficients of terms in a binomial expansioncan be found by using combinations.

What is the probability that 3 coins show heads and 3 show tailswhen 6 coins are tossed?There are 2 possible outcomes that are equally likely: heads (H) and tails (T). The tosses of 6 coins are independent events. When (H � T)6 is expanded, the term containing H3T3,which represents 3 heads and 3 tails, is used to get the desired probability. By the BinomialTheorem the coefficient of H3T3 is C(6, 3).

P(3 heads, 3 tails) � �36!3!!� ��

12��3��

12��3 P(H) � �

12

� and P(T) � �12

�

� �2604�

� �156�

The probability of getting 3 heads and 3 tails is �156� or 0.3125.

Find each probability if a coin is tossed 8 times.

1. P(exactly 5 heads) 2. P(exactly 2 heads)

about 22% about 11%

3. P(even number of heads) 4. P(at least 6 heads)

50% about 14%

Mike guesses on all 10 questions of a true-false test. If the answers true and falseare evenly distributed, find each probability.

5. Mike gets exactly 8 correct answers. 6. Mike gets at most 3 correct answers.

or 0.044 or 0.172

7. A die is tossed 4 times. What is the probability of tossing exactly two sixes?

or 0.11625�

11�

45�

ExampleExample

ExercisesExercises


Binomial Experiments

A binomial experiment is possible if and only if all of these conditions occur.• There are exactly two outcomes for each trial.

Binomial Experiments • There is a fixed number of trials.• The trials are independent.• The probabilities for each trial are the same.

Suppose a coin is weighted so that the probability of getting heads inany one toss is 90%. What is the probability of getting exactly 7 heads in 8 tosses?

The probability of getting heads is �190�, and the probability of getting tails is �1

10�. There are

C(8, 7) ways to choose the 7 heads.

P(7 heads) � C(8, 7)� �7� �1

� 8 �

� 0.38

The probability of getting 7 heads in 8 tosses is about 38%.

1. BASKETBALL For any one foul shot, Derek has a probability of 0.72 of getting the shotin the basket. As part of a practice drill, he shoots 8 shots from the foul line.

a. What is the probability that he gets in exactly 6 foul shots? about 31%b. What is the probability that he gets in at least 6 foul shots? about 60%

2. SCHOOL A teacher is trying to decide whether to have 4 or 5 choices per question onher multiple choice test. She wants to prevent students who just guess from scoring wellon the test.

a. On a 5-question multiple-choice test with 4 choices per question, what is theprobability that a student can score at least 60% by guessing? 10.4%

b. What is the probability that a student can score at least 60% by guessing on a test ofthe same length with 5 choices per question? 5.8%

3. Julie rolls two dice and adds the two numbers.

a. What is the probability that the sum will be divisible by 3? �13

�

b. If she rolls the dice 5 times what is the chance that she will get exactly 3 sums thatare divisible by 3? about 16%

4. SKATING During practice a skater falls 15% of the time when practicing a triple axel.During one practice session he attempts 20 triple axels.

a. What is the probability that he will fall only once? about 14%b. What is the probability that he will fall 4 times? about 18%

97�108

1�10

9�10



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1. P(4 heads) �116� 2. P(0 heads) �

116�

3. P(exactly 3 heads) �14

� 4. P(exactly 2 heads) �38

�

5. P(exactly 1 head) �14

� 6. P(at least 3 heads) �156�

Find each probability if a die is rolled 3 times.

7. P(exactly one 2) �2752� 8. P(exactly two 2s) �

752�

9. P(exactly three 2s) �2116� 10. P(at most one 2) �

2257�

A town that presents a fireworks display during its July 4 celebration found the probability that a family with two or more children will watch the fireworks is �

35�.

If 5 of these families are selected at random, find each probability.

11. P(exactly 3 families watch the fireworks) 12. P(exactly 2 families watch the fireworks)

�261265

� �164245

�

13. P(exactly 5 families watch the fireworks) 14. P(no families watch the fireworks)

�3214235

� �331225�

15. P(at least 4 families watch the fireworks) 16. P(at most 1 family watches the fireworks)

�13015235

� �3217225

�

One section of a standardized English language test has 10 true/false questions.Find each probability when a student guesses at all ten questions.

17. P(exactly 8 correct) �140524� 18. P(exactly 2 correct) �

140524�

19. P(exactly half correct) �26536

� 20. P(all 10 correct) �10

124�

21. P(0 correct) �10

124� 22. P(at least 8 correct) �

1728�



1. P(exactly 3 tails) �156� 2. P(exactly 5 tails) �

332�

3. P(0 tails) �614� 4. P(at least 4 heads) �

1312�

5. P(at least 4 tails) �1312� 6. P(at most 2 tails) �

1312�

The probability of Chris making a free throw is �23�. If she shoots 5 times, find each

probability.

7. P(all missed) �2143� 8. P(all made) �

23423

�

9. P(exactly 2 made) �24403

� 10. P(exactly 1 missed) �28403

�

11. P(at least 3 made) �6841� 12. P(at most 2 made) �

1871�

When Tarin and Sam play a certain board game, the probability that Tarin will win a game is �

34�. If they play 5 games, find each probability.

13. P(Sam wins only once) �1400254

� 14. P(Tarin wins exactly twice) �54152

�

15. P(Sam wins exactly 3 games) �54152

� 16. P(Sam wins at least 1 game) �1708214

�

17. P(Tarin wins at least 3 games) �455192

� 18. P(Tarin wins at most 2 games) �55132

�

19. SAFETY In August 2001, the American Automobile Association reported that 73% ofAmericans use seat belts. In a random selection of 10 Americans in 2001, what is theprobability that exactly half of them use seat belts? Source: AAA about 7.5%

HEALTH For Exercises 20 and 21, use the following information.In 2001, the American Heart Association reported that 50 percent of the Americans whoreceive heart transplants are ages 50–64 and 20 percent are ages 35–49. Source: American Heart Association

20. In a randomly selected group of 10 heart transplant recipients, what is the probabilitythat at least 8 of them are ages 50–64? �

1728�

21. In a randomly selected group of 5 heart transplant recipients, what is the probabilitythat 2 of them are ages 35–49? �

162285

�

Practice (Average)


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Pre-Activity How can you determine whether guessing is worth it?


Suppose you are taking a 50-question multiple-choice test in which thereare 5 answer choices for each question. You are told that no points will bededucted for wrong answers. Should you guess the answers to the questionsyou do not know? Explain your reasoning. Sample answer: Yes; the probability of guessing the right answer to a question is �

15

�, so you have a chance to get some points by guessing, and youhave nothing to lose.

Reading the Lesson1. Indicate whether each of the following is a binomial experiment or not a binomial

experiment. If the experiment is not a binomial experiment, explain why.

a. A fair coin is tossed 10 times and “heads” or “tails” is recorded each time. binomialexperiment

b. A pair of dice is thrown 5 times and the sum of the numbers that come up is recordedeach time. Not a binomial experiment; there are more than two possibleoutcomes for each trial.

c. There are 5 red marbles and 6 blue marbles in a bag. One marble is drawn from thebag and its color recorded. The marble is not put back in the bag. A second marble isdrawn and its color recorded. Not a binomial experiment; the trials are notindependent (or, the probabilities for the two trials are not the same).

d. There are 5 red marbles and 6 blue marbles in a bag. One marble is drawn from thebag and its color recorded. The marble is put back in the bag. A second marble isdrawn and its color recorded. binomial experiment

2. Len randomly guesses the answers to all 6 multiple-choice questions on his chemistrytest. Each question has 5 choices. Which of the following expressions gives theprobability that he will get at least 4 of the answers correct? B

A. P(6, 4)��15��4��

45��2

� P(6, 5)��15��5��

45��1

� P(6, 6)��15��6��

45��0

B. C(6, 4)��15��4��

45��2

� C(6, 5)��15��5��

45��1

� C(6, 6)��15��6��

45��0

C. C(6, 4)��15��2��

45��4

� C(6, 5)��15��1��

45��5

� C(6, 6)��15��0��

45��6

Helping You Remember3. Some students have trouble remembering how to calculate binomial probabilities. What is

an easy way to remember which numbers to put into an expression like C(6, 4)��15��2��

45��4?

Sample answer: The binomial coefficient is C(n, r), where n is thenumber of trials and r is the number of successes. The probability ofsuccess is raised to the r th power and the probability of failure is raisedto the


Misuses of StatisticsStatistics can be misleading. Graphs for a set of data can look very differentfrom one another. Compare the following graphs.

Notice that the two graphs show the same data, but the spacing in thevertical and horizontal scales differs. Scales can be cramped or spread out tomake a graph that gives a certain impression. Which graph would you use togive the impression that the unemployment rate dropped dramatically from1990 to 2000?

Suppose that a car company claims, “75% of people surveyed say that our caris better than the competition.” If four people were asked which car theypreferred and 75% agreed, how many people thought that Our Car wasbetter?

The advertisement was misleading in other ways as well. For example, whowas surveyed—were the people company employees, or impartial buyers?

Suppose an advertiser claims that 90% of all of one brand of car soldin the last 10 years are still on the road.

1. If 10,000 cars were sold, how many are still on the road? 9,000

2. If 1000 cars were sold, how many are still on the road? 900

3. Find an example to show how you think averages could be used in amisleading way. See students’ work.

4. A survey of a large sample of people who own small computers revealedthat 85% of the people thought the instruction manuals should be betterwritten. A manufacturer of small computers claimed that it surveyedmany of the same people and found that all of them liked their manuals.Discuss the possible discrepancy in the results. See students’ work.

U.S. Unemployment Rate

Year

Perc

ent

0 ’90 ’92 ’94 ’96 ’02’98 ’00

8

7

6

5

4

Source: U.S. Department of Labor

U.S. Unemployment Rate

Year

Perc

ent

0 ’90 ’92 ’94 ’96 ’02’98 ’00’91 ’93 ’95 ’97 ’99 ’01

87654

Source: U.S. Department of Labor

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Bias A sample of size n is random (or unbiased) when every possible sample of size n hasan equal chance of being selected. If a sample is biased, then information obtained from itmay not be reliable.

To find out how people in the U.S. feel about mass transit, people ata commuter train station are asked their opinion. Does this situation represent arandom sample?No; the sample includes only people who actually use a mass-transit facility. The sampledoes not include people who ride bikes, drive cars, or walk.

Determine whether each situation would produce a random sample. Write yes orno and explain your answer.

1. asking people in Phoenix, Arizona, about rainfall to determine the average rainfall forthe United States No; it rains less in Phoenix than most places in the U.S.

2. obtaining the names of tree types in North America by surveying all of the U.S. NationalForests Yes; there are National Forests in about every state in the U.S.

3. surveying every tenth person who enters the mall to find out about music preferences inthat part of the country Yes; mall customers should be fairly representativein terms of music tastes.

4. interviewing country club members to determine the average number of televisions perhousehold in the community No; country club members would tend to bemore affluent and thus not a representative sample of the community.

5. surveying all students whose ID numbers end in 4 about their grades and careercounseling needs Yes; ID numbers are probably assigned alphabetically orby some other method not connected to students’ grades or counselingneeds.

6. surveying parents at a day care facility about their preferences for brands of baby foodfor a marketing campaign Yes; choice of a daycare facility would probablynot influence baby food preferences.

7. asking people in a library about the number of magazines to which they subscribe inorder to describe the reading habits of a town No; library visitors tend to readmore than most citizens.

ExampleExample

ExercisesExercises


Margin of Error The margin of sampling error gives a limit on the differencebetween how a sample responds and how the total population would respond.

If the percent of people in a sample responding in a certain way is p and the size of the sample Margin of Error is n, then 95% of the time, the percent of the population responding in that same way will be

between p � ME and p � ME, where ME � 2�.

In a survey of 4500 randomly selected voters, 62% favoredcandidate A. What is the margin of error?

ME � 2� Formula for margin of sampling error

� 2� p � 62% or 0.62, n � 4500

� 0.01447 Use a calculator.

The margin of error is about 1%. This means that there is a 95% chance that the percent ofvoters favoring candidate A is between 62 � 1 or 61% and 62 � 1 or 63%.

The CD that 32% of teenagers surveyed plan to buy next is thelatest from the popular new group BFA. If the margin of error of the survey is 2%,how many teenagers were surveyed?

ME � 2� Formula for margin of sampling error

0.02 � 2� ME � 0.02, p � 0.32

0.01 � � Divide each side by 2.

0.0001 � Square each side.

n � Multiply by n and divide by 0.0001

n � 2176

2176 teenagers were surveyed.

Find the margin of sampling error to the nearest percent.

1. p � 45%, n � 350 2. p � 12%, n � 1500 3. p � 86%, n � 600about 5% about 2% about 3%

4. A study of 50,000 drivers in Indiana, Illinois, and Ohio showed that 68% preferred aspeed limit of 75 mph over 65 mph on highways and country roads. What was themargin of sampling error to the nearest tenth of a percent? about 0.4%

0.32(0.68)��0.0001

0.32(0.68)��n

0.32(0.68)��n

0.32 � (1 � 0.32)��n

p(1 � p)��n

0.62 � (1 � 0.62)��4500

p(1 � p)��n

p(1 � p)��

n


Sampling and Error

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1. calling households at 3:30 P.M. on Tuesday to determine a political candidate’s supportNo; since most registered voters are likely to be at work at this time, thissample would not be representative of all registered voters.

2. polling customers as they exit a sporting goods store about their attitudes about exerciseNo; these customers are likely to value exercise more than those who donot shop at sporting goods stores, who are not represented in this survey.

3. recording the number of sit-ups performed by 15-year old girls in the high schools of alarge school district to determine the fitness of all high-school girls in the districtNo; 15-year old girls may not have the same abilities as 18-year oldseniors, for example.

4. selecting two of a city’s 20 apartment buildings for a survey to determine the desire ofapartment dwellers in the city to own a home No; the residents of the twobuildings selected might, for example, have nicer apartments or be in anicer area of town, and thus would not well represent the desires ofpeople in other buildings.

5. In a large school district, the superintendent of schools interviews two teachers atrandom from each school to determine whether teachers in the district think studentsare assigned too much or too little homework. Yes; since a cross section ofteachers from all levels was selected at random, the sample should wellrepresent the population of teachers in the district.

6. For seven consecutive days, one hour each in the morning, afternoon, and evening, everytenth customer who enters a mall is asked to choose her or his favorite store. Yes;because the sample is chosen over the course of a whole week, duringhours when different consumer groups shop, and because the selectionis systematic, the sample should well represent the general populationthat shops at the mall stores.


7. p � 85%, n � 100 about 7% 8. p � 78%, n � 100 about 8%

9. p � 15%, n � 100 about 7% 10. p � 37%, n � 500 about 4%

11. p � 12%, n � 500 about 3% 12. p � 93%, n � 500 about 2%

13. p � 23%, n � 1000 about 3% 14. p � 56%, n � 1000 about 3%

15. HEALTH In a recent poll of cigarette smokers, 67% of those surveyed said they had triedto quit smoking within the last year. The margin of error was 3%. About how manypeople were surveyed? about 983



1. calling every twentieth registered voter to determine whether people own or rent theirhomes in your community No; registered voters may be more likely to behomeowners, causing the survey to underrepresent renters.

2. predicting local election results by polling people in every twentieth residence in all thedifferent neighborhoods of your community Yes; since all neighborhoods arerepresented proportionally, the views of the community should as awhole should be well represented.

3. to find out why not many students are using the library, a school’s librarian gives aquestionnaire to every tenth student entering the library No; she is polling onlythe students who are coming to the library, and will obtain no input fromthose who aren’t using the library.

4. testing overall performance of tires on interstate highways only No; for overallperformance, tires should be tested on many kinds of surfaces, andunder many types of conditions.

5. selecting every 50th hamburger from a fast-food restaurant chain and determining itsfat content to assess the fat content of hamburgers served in fast-food restaurant chainsthroughout the country No; the selected hamburgers are a random sampleof the hamburgers served in one chain, and may represent the fatcontent for that chain, but will not necessarily represent the fat contentof hamburgers served in other fast-food restaurant chains.

6. assigning all shift workers in a manufacturing plant a unique identification number, andthen placing the numbers in a hat and drawing 30 at random to determine the annualaverage salary of the workers Yes; because the numbers are randomlychosen from among all shift workers, all workers have the same chanceof being selected.





16. COMPUTING According to a poll of 500 teenagers, 43% said that they use a personalcomputer at home. What is the margin of sampling error? about 4%

17. TRUST A survey of 605 people, ages 13–33, shows that 68% trust their parents more thantheir best friends to tell them the truth. What is the margin of sampling error? about 4%

18. PRODUCTIVITY A study by the University of Illinois in 1995 showed an increase inproductivity by 10% of the employees who wore headsets and listened to music of theirchoice while they were working. The margin of sampling error for the study was about7%. How many employees participated in the study? about 76

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Sampling and Error

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Pre-Activity How are opinion polls used in political campaigns?


Do you think the results of the survey about the presidential preferencesdemonstrates that Bush was actually ahead in Florida a month before theelection? If there is not enough information given to determine this, list atleast two questions you would ask about the survey that would help youdetermine the significance of the survey. Sample answer: There is notenough information to tell. 1. How many people were surveyed?2. How was the sample for the survey selected? 3. What is themargin of error for this survey?

Reading the Lesson

1. Determine whether each situation would produce a random sample. Write yes or no andexplain your answer.

a. asking all the customers at five restaurants on the same evening how many times amonth they eat dinner in restaurants to determine how often the average Americaneats dinner in a restaurants No; people surveyed at a restaurant might belikely to eat dinner in restaurants more often than other people.

b. putting the names of all seniors at your high school in a hat and then drawing 20 namesfor a survey to find out where seniors would like to hold their prom Yes; everysenior would have an equal chance of being chosen for the survey.

2. A survey determined that 58% of registered voters in the United States support increasedfederal spending for education. The margin of error for this survey is 4%. Explain in yourown words what this tells you about the actual percentage of registered voters who supportincreased spending for education. Sample answer: There is a 95% chance thatthe actual percentage of voters supporting increased federal spendingfor education is between 54% and 62%.


3. The formula for margin of sampling error may be tricky to remember. A good way to startis to think about the variables that must be included in the formula. What are thesevariables, and what do they represent? What is an easy way to remember which variablegoes in the denominator in the formula? Sample answer: p is the probability ofa certain response and n is the sample size. The larger the sample size,the smaller the margin of error, so n must go in the denominator sincedividing by a larger number gives a smaller number. The square root of asmaller number is a smaller number, and twice the square root of asmaller number is a smaller number.


Shapes of Distribution CurvesGraphs of frequency distributions can be described as either symmetric or skewed.

In a distribution skewed to the right, there are a larger number of highvalues. The long “tail” extends to the right.

In a distribution skewed to the left, there are a larger number of low values.The “tail” extends to the left.

For each of the following, state whether the distribution is symmetricor skewed. If it is skewed, tell whether it is skewed to the right or tothe left.

1. 2. 3.

symmetric skewed to the left skewed to the right

4. 5. 6.

symmetric symmetric skewed to the right

A vertical line above the median divides the area under a frequencycurve in half.

7. Where is the median in a symmetric 8. Where is the median in a skeweddistribution? In the middle of the distribution? To the left of the middle range; it is the same as the mean. if skewed to the right; to the right

of the middle if skewed to the left.

Symmetric Skewed to the Right Skewed to the Left

MedianModeMean Median

Mode Mean

Median

ModeMean

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-912-9

Write the letter for the correct answer in the blank at the right of each question.

1. Carl purchased four new shirts and three new pairs of pants. How many new outfits can he make with these items?A. 12 B. 7 C. 9 D. 81 1.

2. TRIATHALON During training, a triathlete works on biking, swimming,and running times. How many ways can a triathlete choose the order of these activities in a training session?A. 4 B. 9 C. 5 D. 6 2.

3. Evaluate P(7, 2).A. 49 B. 21 C. 42 D. 14 3.

4. Evaluate C(6, 2).A. 30 B. 15 C. 12 D. 36 4.

5. Find the odds of an event occurring, given that the probability of the event

is �131�.

A. 3:11 B. 8:3 C. 8:11 D. 3:8 5.

6. The table and relative-frequency histogram show the distribution of the number of tails when 2 coins are tossed. Find P(T � 2 tails).

A. �14� B. �

12�

C. 1 D. 0 6.

7. A blue die and a red die are tossed. What is the probability that a 6 will appear on both dice?

A. �118�

B. �316�

C. �12� D. �1

11�

7.

8. A jar contains 10 purple marbles and 2 red marbles. If two marbles are chosen at random with no replacement, what is the probability that 2 purple marbles are chosen?

A. �2356�

B. �56� C. �

1252�

D. �15� 8.

9. A bag contains 6 cherry, 8 strawberry, and 9 grape-flavored candies. What is the probability of selecting a cherry or a grape flavored candy?

A. �1253�

B. �1243�

C. �1273�

D. �55249�

9.

10. A die is rolled. What is the probability of rolling a 6 or a number greater than 4?

A. �23� B. �

12� C. �

16� D. �

13� 10.

11. A coin is tossed 5 times. Find P(5 tails).

A. �15� B. �1

10�

C. �116�

D. �312�

11.

Chapter 12 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

1212

T � Tails 0 1 2

Probability �14

� �12

� �14

�210

0

Pro

bab

ility

Tails

12

34

14


Chapter 12 Test, Form 1 (continued)

12. Which measure of central tendency best represents a data set with outliers?A. mode B. mean C. median D. variance 12.

For Questions 13–15, use the data set {10, 12, 12, 14, 22}.

13. Find the mean.A. 17.5 B. 14 C. 70 D. 13 13.

14. Find the variance. Round to the nearest tenth, if necessary.A. 17.6 B. 88 C. 4.2 D. 4 14.

15. Find the standard deviation. Round to the nearest tenth, if necessary.A. 17.6 B. 14.6 C. 4.2 D. 14 15.

16. Classify the data in the table.A. positively skewedB. negatively skewedC. normally distributedD. discrete distribution 16.

17. CAR SALES The mean stay of a car on a lot before being sold is 21 days, with a standard deviation of 3 days. The lengths of stay are normally distributed. What percent of the cars are sold after having been on the lot between 18 and 24 days?A. 95% B. 34% C. 68% D. 5% 17.

18. The probability that a certain team will win a baseball game is �13�. In a

5-game series, what is the probability that the team will win all five games?

A. �115�

B. �2143�

C. �13� D. �2

543�

18.

19. COMMUTERS Which group should be surveyed to determine how people commute to work in order to produce a random sample?A. students in your schoolB. people passing through a toll booth on a given dayC. people in your state whose last name begins with SD. people whose annual income is greater than $1,000,000 19.

20. Find the margin of sampling error when p � 45% and n � 100 if

ME � 2��p(1

n�� p)��.

A. 9% B. 10% C. 5% D. 1% 20.

Bonus If f represents the probability of rolling a 5 and nrepresents the probability of rolling any other number,which term of (f � n)4 � f 4 � 4f 3n � 6f 2n2 � 4fn3 � n4

represents the probability of rolling exactly three 5s in 4 rolls of a die? Find the probability. B:

NAME DATE PERIOD

1212

Amount Spent on LunchLess than $4.00 18%

$4.00–$7.99 47%

$8.00–$11.99 16%

$12.00–$15.99 11%

$16.00 or more 8%

Chapter 12 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. LICENSE PLATES A license plate has one letter (not I or O) followed by five digits. How many different plates are possible?A. 1200 B. 2,400,000 C. 725,760 D. 100,000 1.

2. How many 3-letter identification codes are possible if no letter is repeated?A. 17,576 B. 2600 C. 78 D. 15,600 2.

3. Evaluate P(10, 4).A. 5040 B. 151,200 C. 30,240 D. 210 3.

4. A group has 6 men and 5 women. How many ways can a committee of 3 men and 2 women be formed?A. 200 B. 150 C. 7200 D. 2400 4.

5. The odds that an event will occur are 7:2. What is the probability that the event will occur?

A. �194�

B. �79� C. �

29� D. �

27� 5.

6. Two marbles are chosen at random from a bag containing 3 blue and 2 red marbles. The relative-frequency histogram shows the distribution of the number of red marbles chosen. Find P(2 red).

A. �110�

B. �15�

C. �35� D. �1

30�

6.

7. A red die and a blue die are tossed. What is the probability that the red die shows a 5 and the blue die shows an even number?

A. �316�

B. �118�

C. �112�

D. �23� 7.

8. Tickets are numbered 1 to 50 and are placed in a box. Three tickets are drawn at random without replacement. What is the probability that the numbers are all greater than 35?

A. �120700� B. �5

1630�

C. �130�

D. �78140� 8.

9. From 4 yellow and 9 blue marbles, 3 are selected. What is the probability that all 3 are yellow or all 3 are blue?

A. �1443�

B. �143�

C. �14423�

D. �18443�

9.

10. A card is drawn from a deck of cards. What is the probability of drawing a club or a face card? (Hint: A face card is a jack, queen, or king.)

A. �2552�

B. �133�

C. �2161� D. �1

73�

10.

11. A coin is tossed 5 times. Find P(at least 3 tails).

A. �136�

B. �12� C. �1

56�

D. �35� 11.

1212

2100

Pro

bab

ility

Red

15

25

3512

310

110


Chapter 12 Test, Form 2A (continued)

12. How many different arrangements of the letters of the word radar are possible?A. 120 B. 60 C. 30 D. 480 12.

TEMPERATURES For Questions 13–15, use the data in the table. Round to the nearest tenth, if necessary.

Source: www.weather.com

13. Which measure of central tendency is not a good representation of the data?A. mean B. mode C. median D. middle 13.

14. Find the variance of the temperatures.A. 28.4 B. 5.3 C. 59.3 D. 340.7 14.

15. Find the standard deviation of the temperatures.A. 52�F B. 5.3�F C. 5.6�F D. 28.4�F 15.


17. POTTERY The diameters of pottery bowls are normally distributed. The mean of the diameters is 22 cm and the standard deviation is 2 cm. What percent of the bowls have diameters between 18 and 26 cm?A. 13.5% B. 34% C. 68% D. 95% 17.

18. In a local car lot, �16� of the cars have standard transmissions. Find the

probability that 3 of 4 randomly-selected cars have standard transmissions.

A. �132254�

B. �59� C. �3

524�

D. �12596� 18.

19. A school librarian wants to determine the reading interests of students. A survey of which group would produce a random sample?A. every third student leaving the library on a given dayB. students on the football teamC. every fifth person entering the school in the morningD. seniors planning to attend college 19.

20. HOMEWORK In a survey of 320 students, 32% spent at least 1 hour per night on homework. Find the margin of sampling error.A. 5% B. 21% C. 3% D. 10% 20.

Bonus Write a data set having 7 values that has a median of 24 and a mean of 20. B:

NAME DATE PERIOD

1212

Record Low Temperatures in Honolulu, HI (�F)Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

52 53 55 56 60 65 66 67 66 61 57 54

Age of Population of Iowa in 2000Age Number of People

0–24 978,875

25–44 795,499

45–64 644,861

65–84 357,074

Over 84 45,848

Chapter 12 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent

Write the letter for the correct answer in the blank at the right of each question. 1. An ice cream store has 31 flavors of ice cream and 10 toppings. A regular

sundae has one flavor of ice cream, one topping, and comes with or without whipped cream. How many different ice cream sundaes can be ordered?A. 310 B. 372 C. 620 D. 82 1.

2. How many 5-digit codes are possible if 0 cannot be used and no digit can be repeated?A. 15,120 B. 45 C. 30,240 D. 59,049 2.

3. A clown has 7 balloons, each a different color. There are 5 children.How many ways can the clown give each child a balloon?A. 21 B. 5040 C. 42 D. 2520 3.

4. Evaluate C(13, 9).A. 17,160 B. 715 C. 259,459,200 D. 117 4.

5. The probability that an event will occur is �1151�. What are the odds that the

event will occur?A. 15:11 B. 11:15 C. 4:11 D. 11:4 5.

6. Two marbles are chosen at random from a bag containing 3 blue and 2 red marbles. The relative-frequency histogram shows the distribution of the number of red marbles chosen. Find P(0 red).

A. 0 B. �185�

C. �115�

D. �25� 6.

7. A red die and a blue die are tossed. What is the probability that the red die shows a 3 and the blue die shows a number greater than 3?

A. �110�

B. �15� C. �1

30�

D. �35� 7.

8. Tickets are numbered 1 to 50 and placed in a box. Three tickets are drawn at random without replacement. What is the probability that their numbers are all greater than 25?

A. �18� B. �1

2936�

C. �66295�

D. �12� 8.

9. From 4 yellow and 8 blue marbles, 3 are selected. What is the probability that all three are yellow or all three are blue?

A. �131�

B. �515�

C. �1545�

D. �2320�

9.

10. A card is drawn from a standard deck of cards. What is P(heart or a 6)?

A. �296�

B. �1572�

C. �14� D. �1

43�

10.

11. A coin is tossed 5 times. Find P(at most 4 tails).

A. �136�

B. �1136�

C. �312�

D. �3312�

11.

1212

2100

Pro

bab

ility

Red

15

25

3512

310

110


Chapter 12 Test, Form 2B (continued)

12. How many different arrangements of the letters of the word doodle are possible?A. 180 B. 720 C. 15 D. 90 12.

TEMPERATURES For Questions 13–15, use the data in the table. Round to the nearest tenth, if necessary.


13. Which measure of central tendency is not a good representation of the data?A. middle B. median C. mode D. mean 13.

14. Find the variance of the temperatures.A. 4366.2 B. 64.6 C. 2342.9 D. 195.2 14.

15. Find the standard deviation of the temperatures.A. 14.6�F B. 14.0�F C. 63.0�F D. 64.6�F 15.


17. For 2000 patients, blood-clotting time was normally distributed with a mean of 8 seconds and a standard deviation of 3 seconds. What percent had blood-clotting times between 5 and 11 seconds?A. 68% B. 34% C. 49.5% D. 47.5% 17.

18. During a sale, �16� of the CD prices are reduced. Find the probability that 2

of 4 randomly-selected CDs have reduced prices.

A. �356�

B. �122596� C. �2

2156�

D. �2516�

18.

19. A music teacher wants to determine the music preferences of students.A survey of which group would produce a random sample?A. students in the school bandB. students attending the annual jazz concertC. students in every odd-numbered homeroomD. every other player on the baseball roster 19.

20. ELECTIONS In an election poll, 56% of 400 voters chose a certain candidate. Find the margin of sampling error.A. 5% B. 2% C. 4% D. 7% 20.

Bonus Write a data set having 7 data values that has a median of 20 and a mean of 24. B:

NAME DATE PERIOD

1212

Record High Temperatures in Anchorage, Alaska (�F)Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

50 48 51 65 77 85 82 82 73 61 53 48

Source: Census 2000

Age of Population of Rhode Island in 2000

Age Number of People

0–14 206,423

15–34 265,778

35–54 308,946

55–74 159,092

over 74 69,264

Chapter 12 Test, Form 2C


1. BELTS A clothing store sells belts in 3 colors, 4 designs, and 1.6 sizes. How many different belts are available?

2. Five children stand in a line to play a game. How many 2.different ways can the children be arranged?

3. CROSS-COUNTRY Twelve runners are in a cross-country 3.race. How many different ways can they finish first, second,and third?

4. Five cheerleaders will be chosen from a group of 15 students. 4.How many different cheerleading squads can be formed?

5. The odds of an event occurring are 4 to 7. What is the 5.probability that the event will occur?

6. Two socks are chosen at random from 6.a drawer containing 6 black and 3 blue socks. The table and relative-frequency histogram show the distribution of the number of black socks chosen. Find P(B � 2).

7. A die is rolled three times. What is P(no 5s)? 7.

8. Two cards are drawn from a standard deck of 52 cards 8.without replacement. Find the probability that the first card is an ace and the second is a 2.

9. From a group of 6 men and 8 women, a committee of 3 is 9.selected. Find the probability that all 3 are men or all 3 are women.

10. Each of the numbers 1 to 25 is written on a card and placed 10.in a bag. If one card is drawn at random, what is the probability that it is a multiple of 4 or a multiple of 5?

11. Seven coins are tossed. Find P(at least 6 tails). 11.

12. If the probability of rain in a certain city is �18� on any given 12.

day, find the probability that rain will fall on exactly one day of a three-day visit to the city.

13. How many different arrangements of the letters in the word 13.ILLINOIS are possible?

NAME DATE PERIOD

SCORE 1212

Ass

essm

ent

B � Black 0 1 2

Probability �112� �

12

� �152� 210

0

Pro

bab

ility

Black

16

13

125

12

14

112


Chapter 12 Test, Form 2C (continued)

TEMPERATURES For Questions 14–16, use the data in the table.Round to the nearest tenth, if necessary.


14. If you were a member of the Chamber of Commerce, which 14.measure of central tendency would you use to convince someone that Memphis has a comfortable climate? Explain.

15. Find the variance of the temperatures. 15.

16. Find the standard deviation of the temperatures. 16.

17. EDUCATION Determine whether the data in the table is 17.positively skewed, negatively skewed, or normally distributed.

Source: www.census.gov

18. COLLEGE ENTRANCE EXAM The scores on a standardized 18.college entrance examination are found to be normally distributed. The mean is 85 and the standard deviation is 11.What percent scored between 85 and 107?

19. Determine whether the situation would produce a random 19.sample and explain your answer: surveying your class to determine the most-admired person in the United States by people your age.

20. In a sample of 120 small business owners, 64% said they 20.preferred a certain company for office supplies. Find the margin of sampling error.

Bonus Student test grades were normally distributed, and B:grades between 62 and 86 were within three standard deviations of the mean. Find the mean and standard deviation of the set of grades.

NAME DATE PERIOD

1212

Educational Attainment in Georgia for persons over 25 years of age, as of 2000

Less than 9th grade 484,000

9th to 12th grade, no diploma 686,000

High school graduate 1,193,000

Some college or associate degree 884,000

Bachelor’s degree 520,000

Graduate or professional degree 258,000

Record High Temperatures in Memphis, TN (�F)Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

78 81 85 94 99 104 108 105 103 95 85 81

Chapter 12 Test, Form 2D


1. A store sells T-shirts in 7 colors, 5 designs, and 3 sizes. How 1.many different T-shirts are available?

2. Marva needs to mow the lawn, pay her bills, walk the dog, 2.and return a phone call. How many ways can she choose to order her tasks?

3. How many different arrangements of three coins can be 3.made if you have a penny, a nickel, a dime, a quarter, and a silver dollar?

4. How many different 5-player basketball teams can be 4.formed from a group of 12 people?

5. If the probability that an event will occur is �152�

, what are 5.

the odds that it will occur?

6. Two socks are chosen at random from 6.a drawer containing 4 black and 3 blue socks. The table and relative-frequency histogram show the distribution of the number of blue socks chosen. Find P(B � 1).

7. A die is rolled three times. What is P(three 5s)? 7.

8. Two cards are drawn from a standard deck of 52 cards 8.without replacement. Find the probability that both cards are aces.

9. From a group of 7 men and 5 women, a 4-person committee 9.is chosen. What is the probability that all 4 are men or all 4 are women?

10. Each of the numbers 1 to 20 is written on a card and placed 10.in a bag. If one card is drawn at random, what is the probability that it is a multiple of 3 or a multiple of 5?

11. Eight coins are tossed. Find P(at least 7 heads). 11.

12. If the probability of rain in a certain city is �25� on any given 12.

day, find the probability that rain will fall on exactly one day of a three-day visit to the city.

NAME DATE PERIOD

SCORE 1212

Ass

essm

ent

B � Blue 0 1 2

Probability �27

� �47

� �17

�

2100

Pro

bab

ility

Blue

17

27

37

47


Chapter 12 Test, Form 2D (continued)

13. How many different arrangements of the letters in the word INDIANA are possible? 13.

14. The sales prices of several cars on a used car lot are 14.$18,900; $20,500; $29,900; $19,800; and $21,750. Which measure of central tendency best represents the data? Explain.

For Questions 15 and 16, use the data in the table that shows average precipitation in Grand Junction,Colorado. Round to the nearest hundredth, if necessary.


15. Find the variance of the data. 15.

16. Find the standard deviation of the data. 16.

17. Determine whether the data 17.in the table is positively skewed, negatively skewed,or normally distributed.

18. COLLEGE ENTRANCE EXAM The scores on a 18.standardized college entrance examination are found to be normally distributed. The mean is 78 and the standard deviation is 13. What percent scored between 52 and 78?

19. Determine whether the situation would produce a random 19.sample and explain your answer: surveying persons with library cards to determine if a city should raise taxes to pay for a new library.

20. In a survey of 60 customers in a supermarket, 40% expect to 20.use the express line. What is the margin of sampling error?

Bonus Student test grades were normally distributed, and B:grades between 68 and 86 were within three standard deviations of the mean. Find the mean and standard deviation of the set of grades.

NAME DATE PERIOD

1212

Age of Population of Florida in 2000

Age Number of People

0–24 4,870,160

25–44 4,442,638

45–64 3,581,676

65–84 2,449,573

Over 84 269,388

Average Precipitation in Grand Junction, CO (in.)Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.6 0.5 0.9 0.8 0.9 0.5 0.7 0.8 0.8 1.0 0.7 0.6

Source: Census 2000

Chapter 12 Test, Form 3


1. Each day, Jonathan chooses one of six routes to work. How 1.many different ways can Jonathan get to work over a five-day period?

2. How many different ways can 9 entertainers appear on an 2.awards show if the guest of honor must appear first or last?

3. How many ways can you select 4 pizza toppings from a total 3.of 8 toppings? Is this a permutation or a combination? Explain.

4. Evaluate C(13, 5) � C(9, 4). 4.

5. A coin purse contains 4 pennies, 5 nickels, and 8 dimes. 5.Three coins are selected at random. Find the probability of selecting one coin of each type.

6. Three students are selected at 6.random from a group of 4 males and 6 females. The table and relative-frequency histogram show the distribution of the number of males chosen. Find P(two females).

7. A die is rolled four times. Find P(four of the same number). 7.

8. Four cards are drawn from a standard deck of 52 cards 8.without replacement. Find the probability that the first card is a heart, the second is a club, and the third and fourth are diamonds.

9. From a group of 8 men and 10 women, a committee of 5 is 9.to be selected at random. Find P(at least 3 men).

10. Two cards are drawn from a standard deck of cards. Find 10.P(both black or both 9s).

11. How many ways can 4 basketball shoes, 2 tennis shoes, and 11.5 running shoes be arranged on a shelf if the shoes are grouped according to type?

NAME DATE PERIOD

SCORE 1212

Ass

essm

ent

M � Male 0 1 2

Probability �16

� �12

� �130� �

310�

3

32100

Pro

bab

ility

Males

110

730

1130

1330

12

310

16

130


Chapter 12 Test, Form 3 (continued)

TAXES For Questions 12–15, use the data in the table that shows the per capita taxes, in dollars, in the 10 states listed. Round to the nearest cent, if necessary.

Source: World Almanac

12. Which measure of central tendency might a realtor in 12.Maryland use to convince a client that the per capita taxes were reasonable? Explain.

13. Find the variance of the taxes. 13.

14. Find the standard deviation of the taxes. 14.

15. Determine whether the data in the table is positively 15.skewed, negatively skewed, or normally distributed.

16. IQ TESTS Scores on an IQ test are normally distributed. 16.The mean is 100 and the standard deviation is 15. If 6000 people took the test, how many of them scored between 85 and 130?

17. Find P(at least four 4s) if a die is rolled 6 times. 17.

18. In a certain city in June, the probability that the 18.temperature will rise above 80�F is 0.7. For the first 8 days,what is P(temperature will rise above 80�F exactly 3 times)?Round to the nearest hundredth.

19. Determine whether the situation would produce a random 19.sample and explain your answer: surveying town residents whose license number ends in 5 to determine whether to increase taxes to pay for road repair.

20. PETS In a survey of pet owners, 68% preferred dogs to any 20.other kind of pet. The margin of sampling error was 5%.How many people were surveyed?

Bonus 20% of the students in a high school were surveyed to B:determine their favorite pizza topping. If 43% of those surveyed responded “pepperoni,” and the margin of sampling error was 6.2%, how many students attend the high school?

NAME DATE PERIOD

1212

State Taxes State TaxesArizona 1489 Iowa 1678

California 2073 Maine 1905

Colorado 1483 Maryland 1790

Delaware 2665 Michigan 2161

Illinois 1641 Missouri 1512

Chapter 12 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.

1. Kathy, Alma, and Steven are working on a group quiz. Onequestion is as follows.Two dice are rolled. Find the probability that the first die is a 5 ora 6, and the second die is an even number.All three students agree to let A represent rolling a 5 or a 6, and B represent rolling an even number. But Kathy argues that the

solution is P(A) � P(B) � �26� � �

36� � �

56�, Alma feels certain that the

solution should be P(A) � P(B) � �26� � �

36� � �3

66�

� �16�, and Steven is

convinced that the correct solution is P(A) � P(B) � P(A and B) �

�26� � �

36� � �

16� � �

46� � �

23�.

a. Which student, if any, is correct? Explain your reasoning.b. For one of the incorrect solutions above, write a probability

problem for which that solution would be correct.2. a. One day, your math teacher, Mr. Butler, looks at your exam

scores and informs you that your score distribution isnegatively skewed. How do you feel about this news? Explainyour reasoning.

b. The next day, Mr. Butler announces that the class scores on thelast exam were normally distributed, that scores between 56and 98 fell within three standard deviations of the mean, andthat students whose scores fell within one standard deviationof the mean would earn a grade of C on the exam. Explain howto estimate the mean score, the standard deviation of the classscores, and the range of grades for which a student would earna grade of C. Determine the indicated values.

3. Greg and Jacqui are planning a dinner party for 6 guests. Afterdinner, they plan to separate into two teams to play charades.a. Explain how you could determine the number of different

possible arrangements of guests and hosts into two teams.Include in your explanation whether the formula

P(n, r) � �(n �n!

r)!� or the formula C(n, r) � �(n �n!

r)!r!� would be

helpful in determining the number of arrangements. Explainyour reasoning and determine the number of arrangementsthat are possible.

b. Would the number of possible arrangements change if Gregand Jacqui decided that they should be on different teams? If so, how many arrangements would be possible under thoseconditions? Explain your reasoning.

NAME DATE PERIOD

SCORE 1212

Ass

essm

ent


Chapter 12 Vocabulary Test/Review

Write whether each sentence is true or false. If false, replace the underlined word or words to make a true sentence.

1. A selection of objects in which order is not important is 1.called a permutation.

2. The graph of a normal distribution is a bell curve. 2.

3. Range, variance, and odds are measures of the spread of a 3.set of data.

4. Probability distributions with a finite number of possible 4.values are called continuous probability distributions.

5. Tossing a coin ten times is an example of a 5.linear permutation.

6. A relative-frequency distribution is a graph of a 6.probability distribution.

7. Two or more choices for which the result of one choice does 7.not affect the result of another are called independent events.

8. Events that consist of two or more simple events are called 8.dependent events.

9. The mean, the median, and the mode are 9.measures of variation.

10. A curve or histogram that is not symmetric represents a(n) 10.unbiased sample.

In your own words—Define each term.

11. mutually exclusive events

12. random sample

area diagrambinomial experimentcombinationcompound eventcontinuous probability distribution

dependent eventsdiscrete probability distributions

event

failureFundamental Counting Principle

inclusive eventsindependent eventslinear permutationmargin of sampling errormeasure of central tendency

measure of variation

mutually exclusive eventsnormal distributionoddsoutcomepermutationprobabilityprobability distributionrandomrandom variable

relative-frequency histogram

sample spacesimple eventskewed distributionstandard deviationsuccessunbiased samplevariance

NAME DATE PERIOD

SCORE 1212

Chapter 12 Quiz (Lessons 12–1 through 12–3)

1212


1. Standardized Test Practice Lisa selects a car from 4 models. Each model comes in 5 colors. How many different ways can she select a car?A. 24 B. 16 C. 9 D. 20 1.

2. How many four-digit codes are possible if no digit may be 2.used more than once?

3. A group of 3 women and 1 man is chosen from 7 women and 3.5 men. Does this involve a permutation or a combination? Find the number of different groups that can be formed.

4. Cards are numbered 1 through 20. Find the probability that 4.a card drawn at random will contain a number greater than 11. Then find the odds that a number greater than 11 is drawn.

5. Two marbles are 5.chosen at random from a bag containing 4 red and 3 blue marbles. The table and relative-frequency histogram show the distribution of the number of red marbles chosen. Find P(R � 2).

NAME DATE PERIOD

SCORE

Chapter 12 Quiz (Lessons 12–4 and 12–5)

1. A pair of dice is thrown. What is the probability that both 1.dice show a number less than 5?

For Questions 2 and 3, consider a bag that contains 8 red marbles, 5 white marbles, and 2 blue marbles.

2. If 3 marbles are selected in succession with replacement, 2.what is the probability that the marbles are white, blue,and red in that order?

3. If 3 marbles are selected in succession without replacement, 3.what is the probability that the marbles are white, blue, and red in that order?

4. Janet has 3 dimes and 6 nickels in her pocket. She selects 4.3 coins without replacement. What is the probability that she selects all dimes or all nickels?

5. A card is drawn from a standard deck of 52 playing cards. 5.What is the probability that a heart or face card is drawn? (Hint: A face card is a jack, queen, or king.)

NAME DATE PERIOD

SCORE 1212

Ass

essm

ent

R � Red 0 1 2

Probability �17

� �47

� �27

�

2100

Pro

bab

ility

Red

17

27

37

47


1. Find the variance of the data set {13, 16, 17, 18, 16, 12, 14, 1.12}. Round to the nearest hundredth, if necessary.

2. Find the standard deviation for the data in Question 1. 2.Round to the nearest hundredth, if necessary.

3. Determine if the 3.data in the table appear to be positively or negatively skewed or normally distributed.

The times a group of high school students wake up on weekday mornings was found to be normally distributed.The mean wake-up time was 6:45 A.M. and the times had a standard deviation of 15 minutes.

4. What percent of the students would you expect to wake up 4.between 6:30 A.M. and 7:00 A.M.?

5. If 400 students were surveyed, how many would you expect 5.to wake up between 6:00 A.M. and 7:30 A.M.?


For Questions 1 and 2, find each probability if a die is rolled 3 times.

1. P(exactly two 4s) 2. P(at most two 4s)

3. A batter’s probability of getting a hit is �13�. In his next 3.

5 times at bat, what is the probability that he will get at least 4 hits?

4. Determine whether the situation would produce a random 4.sample and explain your answer: surveying students on the basketball team to determine the favorite sport of students in your school.

5. In a survey of 50 people, 80% read a newspaper at least 5.once per week. Find the margin of sampling error.

NAME DATE PERIOD

SCORE


1212

NAME DATE PERIOD

SCORE

1212

Family Income and Benefits in 2000

Income and Benefits Number of Families (in millions)

less than $50,000 35.8

$50,000–$99,999 24.3

$100,000–$149,999 6.9

$150,000–$199,999 2.0

$200,000 or more 1.9

1.

2.

Chapter 12 Mid-Chapter Test (Lessons 12–1 through 12–4)



1. A company manufactures bicycles in 8 different styles. Each style comes in 7 different colors. How many different bicycles does the company make?A. 64 B. 49 C. 56 D. 15 1.

2. How many ways can 6 children form a line to use the drinking fountain?A. 120 B. 720 C. 36 D. 30 2.

3. Find P(9, 4).A. 126 B. 15,120 C. 36 D. 3024 3.

4. Find C(10, 8).A. 1,814,400 B. 80 C. 90 D. 45 4.

5. The probability that an event will occur is �27�. What are the odds that the

event will occur?A. 2:5 B. 5:2 C. 2:7 D. 2:9 5.

6. A die is rolled twice. Find P(4, then 5).

A. �310�

B. �13� C. �3

16�

D. �59� 6.

7. A jar contains 7 red, 8 blue, and 4 green marbles. What is 7.the probability of choosing 3 blue marbles in a row, if no replacement occurs?

8. A stained glass window has 25 blue pieces and 20 red pieces. 8.If 2 pieces are selected at random, what is P(2 red or 2 blue)?

9. SOCCER On the all-state soccer team, 5 of the 8 players 9.from the North Region are seniors, and 8 of the 12 players from the South Region are seniors. What is the probability that a randomly-selected student is a senior or is a student from the North Region?

10. How many different arrangements of three folders can be 10.made if you have one green, one red, one blue, and one black folder?

11. A bag contains 6 red dice and 10 blue dice. Two dice are 11.selected at random. Find the probability of selecting one red die and one blue die.

12. How many different groups of 3 students can be formed if 12.there are 20 students in the class?

Part II

Part I

NAME DATE PERIOD

SCORE 1212

Ass

essm

ent


Chapter 12 Cumulative Review (Chapters 1–12)

1. The vertices of RST are R(�1, �3), S(2, 4), and T(�4, 3).The triangle is reflected over the line y � x. Find the coordinates of R�S�T�. (Lesson 4-4) 1.

2. Simplify (x2 � 3) � (4x2 � 5x � 9). (Lesson 5-2) 2.

3. Determine whether f(x) � 5x � 8 and g(x) � x � �85� are 3.

inverse functions. (Lesson 7-8)

For Questions 4 and 5, graph the function or equation.

4. (x � 5)2 � y2 � 9 (Lesson 8-3)

5. f(x) � ��(x �2

1)2� (Lesson 9-3)

6. Solve �ww� 3� � w � �w �

33�. Check your solution(s). (Lesson 9-6)

7. Write the equation log1000 �110�

� ��13� in exponential form.

(Lesson 10-2)

8. A savings account deposit of $500 is to earn 5.7% interest.After how many years will the investment be worth $750? Use y � a(1 � r)t and round to the nearest tenth. (Lesson 10-6)

9. Find the three arithmetic means between 5 and �7. (Lesson 11-1) 9.

10. Find four geometric means between 27 and �19�. (Lesson 11-3) 10.

11. Write 0.6�2�7� as a fraction. (Lesson 11-5) 11.

12. How many different ways can the letters of the word 12.PERMUTATION be arranged? (Lesson 12-2)

13. Three cards are drawn from a standard deck of cards 13.without replacement. Find the probability of drawing a king, a queen, and another king in that order. (Lesson 12-4)

14. The scores on an algebra test are found to be normally 14.distributed. The mean is 72 and the standard deviation is 8.What percent scored between 72 and 88? (Lesson 12-7)

NAME DATE PERIOD

1212

4.

5.

6.

7.

8.

xO

f (x )

y

xO

Standardized Test Practice (Chapters 1–12)


1. What is the seventh term in the sequence ��12�, �

14�, ��

18�, �1

16�

, …?

A. �614�

B. ��614�

C. �1128�

D. ��1128�

1.

2. In the correctly completed addition problem shown,� and � are nonzero digits. What number does �represent?E. 5 F. 7G. 4 H. 2 2.

3. The volume of a rectangular box is 405. The length, width,and height of the box are in the ratio 5 : 3 : 1. What is the total surface area of the box?A. 414 B. 27 C. 54 D. 324 3.

4. In the figure shown, what is the value of r?E. 10 F. 14G. 70 H. 7 4.

5. David is twice as old as his sister, Jennifer. Three years ago,David was three times as old as Jennifer. How old is David now?A. 12 B. 9 C. 6 D. 3 5.

6. If a � 0 and b 0, which of the following statements must be true?I. ab � 0 II. b � a 0 III. ac � bc

E. I, II, and III F. I onlyG. I and II only H. II and III only 6.

7. If (x � y)2 � 200 and x2 � y2 � 50, what is the value of xy?A. �75 B. 75 C. 150 D. �150 7.

8. What is 25% of 20% of �34�?

E. 0.375 F. 3.75 G. 0.00375 H. 0.0375 8.

9. What is the sum of all composite numbers between 1 and 15?A. 120 B. 59 C. 78 D. 63 9.

10. If (x � y)2 � 8 and (x � y)2 � 4, what is xy?E. 0 F. 1 G. 2 H. 3 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

1212

Ass

essm

ent

10r � 4t �

7t �

5r �

� 5* �

8 1� 1 �

1 6 0

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.


Standardized Test Practice (continued)

11. Alejandra has been saving to purchase a VCR. 11. 12.The model she wants is priced at $180, on which she will be required to pay 5% sales tax. She has already saved $53. If Alejandra earns $8.50 per hour after all payroll deductions have been made, for how many hours will she need to work in order to have enough money to purchase the VCR?

12. If 43x�2 � 256, what is the value of 32x�1?

13. In the figure at the right, 13. 14.quadrilaterals ABCDand RSTU are similar.What is the value of n?

14. If the average of a and bis 87, the average of aand c is 73, and the average of b and c is 50,what is the average of a, b, and c?

Column A Column B15. d � 4 � 0 15.

16. y � �1, z 5 16.

17. h 0 17.

4h3 � 1�4h4

h� h�

DCBA

��zy

� � 1�z �

zy

�

DCBA

d33d

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Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

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.

99 9 987654321

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NAME DATE PERIOD

1212

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

D C

A

B

14 U

RS

T4

9

n

A

D

C

B

Unit 4 Test (Chapters 11–12)


1. Find the next four terms of the arithmetic sequence 1.4, 10, 16, … .

2. Find the three arithmetic means between 21 and 13. 2.

3. Find Sn for the arithmetic series in which a1 � �11, an � 13, 3.and n � 7.

4. Find the next two terms of the geometric sequence 4.6250, 5000, 4000, … .

5. Find four geometric means between 4096 and 972. 5.

6. Find the sum of a geometric series for which a1 � 1, r � 2, 6.and n � 6.

7. Find a1 in a geometric series for which Sn � 189, r � �12

�, and 7.an � 3.

8. Find the sum of the infinite geometric series 8.36 � 24 � 16 � …, if it exists.

9. Write 0.7�3�5� as a fraction. 9.

10. Find the first five terms of the sequence for which a1 � 5 10.and an�1 � 3an � 1.

11. Find the first three iterates x1, x2, x3 of f(x) � 2x � 5 for 11.an initial value of x0 � 3.

12. Use the Binomial Theorem to find the fifth term in the 12.expansion of (2x � 3y)5.

13. Prove that the statement �15� � �5

12� � �5

13� � … � �5

1n� � �

14��1 � �5

1n�� 13.

is true for all positive integers n. Write your proof on a separate piece of paper.

14. Find a counterexample to the statement 4n � 1 is divisible 14.by 5.

15. A scout troop will prepare trail mix for their next hike. 15.They have decided to mix one type of nut, one type of dried fruit, and one type of granola. The local store carries 8 types of nuts, 6 types of dried fruit, and 5 types of granola. How many different trail mixes are possible?

16. Students are given a list of ten vocabulary words to learn. In 16.how many ways could four of the words be listed on a test?

NAME DATE PERIOD

SCORE

Ass

essm

ent


Unit 4 Test (continued)(Chapters 11–12)

17. Evaluate C(12, 10). 17.

18. If the probability that an event will occur is �183�, what are 18.

the odds that it will occur?

19. A red die and a blue die are tossed. What is the probability 19.that the red die shows an odd number and the blue die shows a 1 or 2?

20. From a group of 6 men and 4 women, a committee of 3 is to 20.be selected at random. Find P(at least 2 women).

21. Two cards are drawn from a standard deck of cards. Find the 21.probability that a king or a red card is drawn.

For Questions 22 and 23, use the data in the table that shows the number of public secondary schools in eight eastern states in the fall of 1998.

22. Find the mean, median, 22.mode, and standard deviation of the data.Round to the nearest hundredth, if necessary.

23. Determine whether the 23.data in the table appear to be positively skewed,negatively skewed, or normally distributed.

24. The time a group of high school students arrive home from 24.school each day was found to be normally distributed. The mean time was 3:15 P.M. and the times had a standard deviation of 15 minutes. What is the probability that a student chosen at random arrives home from school before 2:30 P.M.?

25. During a clothing sale, �14� of the store merchandise is reduced 25.

in price. Find the probability that 3 of 5 randomly-selected shirts have reduced prices.

26. Determine whether the situation would produce a random 26.sample and explain your answer: surveying people at a concert to determine their favorite local radio station.

NAME DATE PERIOD

StateNumber of Public

Secondary SchoolsFlorida 456

Georgia 306

Maine 160

Massachusetts 363

North Carolina 376

New York 935

Rhode Island 54

Virginia 349

Source: World Almanac

Standardized Test PracticeStudent Record Sheet (Use with pages 694–695 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

1212

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

Also enter your answer by writing each number or symbol in a box. Then fill inthe corresponding oval for that number or symbol.

11 13 15

12 14

Select the best answer from the choices given and fill in the corresponding oval.

16 18 20

17 19 21 DCBADCBADCBA

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Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 12-1)

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th

e sh

elf?

24

4.H

ow m

any

lice

nse

pla

tes

con

sist

ing

of t

hre

e le

tter

s fo

llow

ed b

y th

ree

nu

mbe

rs a

repo

ssib

le w

hen

no

repe

titi

on i

s al

low

ed?

11,2

32,0

00

5.S

ixte

en t

eam

s ar

e co

mpe

tin

g in

a s

occe

r m

atch

.Gol

d,si

lver

,an

d br

onze

med

als

wil

l be

awar

ded

to t

he t

op t

hree

fin

ishe

rs.I

n ho

w m

any

way

s ca

n th

e m

edal

s be

aw

arde

d?33

60

6.In

a w

ord-

buil

din

g ga

me

each

pla

yer

pick

s 7

lett

er t

iles

.If

Juli

o’s

lett

ers

are

all

diff

eren

t,h

ow m

any

3-le

tter

com

bin

atio

ns

can

he

mak

e ou

t of

his

7 l

ette

rs?

210

7.T

he e

dito

r ha

s ac

cept

ed 6

art

icle

s fo

r th

e ne

ws

lett

er.I

n ho

w m

any

way

s ca

n th

e 6

arti

cles

be o

rder

ed?

720

8.T

her

e ar

e 10

on

e-h

our

wor

ksh

ops

sch

edu

led

for

the

open

hou

se a

t th

e gr

een

hou

se.

Th

ere

is o

nly

on

e co

nfe

ren

ce r

oom

ava

ilab

le.I

n h

ow m

any

way

s ca

n t

he

wor

ksh

ops

beor

dere

d?3,

628,

800

9.T

he

top

5 ru

nn

ers

at t

he

cros

s-co

un

try

mee

t w

ill

rece

ive

trop

hie

s.If

th

ere

are

22 r

un

ner

sin

th

e ra

ce,i

n h

ow m

any

way

s ca

n t

he

trop

hie

s be

aw

arde

d?3,

160,

080

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Th

e C

ou

nti

ng

Pri

nci

ple

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

____

_

12-1

12-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Th

e C

ou

nti

ng

Pri

nci

ple

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

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12-1

12-1

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Lesson 12-1

Sta

te w

het

her

th

e ev

ents

are

in

dep

end

ent

or d

epen

den

t.

1.fi

nis

hin

g in

fir

st,s

econ

d,or

th

ird

plac

e in

a t

en-p

erso

n r

ace

dep

end

ent

2.ch

oosi

ng

a pi

zza

size

an

d a

topp

ing

for

the

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ain

dep

end

ent

3.S

even

ty-f

ive

raff

le t

icke

ts a

re p

lace

d in

a ja

r.T

hre

e ti

cket

s ar

e th

en s

elec

ted,

one

afte

rth

e ot

her

,wit

hou

t re

plac

ing

a ti

cket

aft

er i

t is

ch

osen

.d

epen

den

t

4.T

he

232

mem

bers

of

the

fres

hm

an c

lass

all

vot

e by

sec

ret

ball

ot f

or t

he

clas

sre

pres

enta

tive

to

the

Stu

den

t S

enat

e.in

dep

end

ent

Sol

ve e

ach

pro

ble

m.

5.A

su

rvey

ing

firm

pla

ns

to b

uy

a co

lor

prin

ter

for

prin

tin

g it

s m

aps.

It h

as n

arro

wed

its

choi

ce t

o on

e of

th

ree

mod

els.

Eac

h o

f th

e m

odel

s is

ava

ilab

le w

ith

eit

her

32

meg

abyt

esof

ran

dom

acc

ess

mem

ory

(RA

M),

64 m

egab

ytes

of

RA

M,o

r 12

8 m

egab

ytes

of

RA

M.

Fro

m h

ow m

any

com

bin

atio

ns

of m

odel

s an

d R

AM

doe

s th

e fi

rm h

ave

to c

hoo

se?

9

6.H

ow m

any

arra

nge

men

ts o

f th

ree

lett

ers

can

be

form

ed f

rom

th

e le

tter

s of

th

e w

ord

MA

TH

if a

ny

lett

er w

ill

not

be

use

d m

ore

than

on

ce?

24

7.A

llan

is

play

ing

the

role

of

Oli

ver

in h

is s

choo

l’s p

rodu

ctio

n o

f O

live

r T

wis

t.T

he

war

drob

e cr

ew h

as p

rese

nte

d A

llan

wit

h 5

pai

rs o

f pa

nts

an

d 4

shir

ts t

hat

he

can

wea

r.H

ow m

any

poss

ible

cos

tum

es c

onsi

stin

g of

a p

air

of p

ants

an

d a

shir

t do

es A

llan

hav

e to

choo

se f

rom

?20

8.T

he 1

0-m

embe

r st

eeri

ng c

omm

itte

e th

at i

s pr

epar

ing

a st

udy

of t

he p

ubli

c tr

ansp

orta

tion

nee

ds o

f it

s to

wn

wil

l se

lect

a c

hai

rper

son

,vic

e-ch

airp

erso

n,a

nd

secr

etar

y fr

om t

he

com

mit

tee.

No

pers

on c

an s

erve

in

mor

e th

an o

ne

posi

tion

.In

how

man

y w

ays

can

th

eth

ree

posi

tion

s be

fil

led?

720

9.Je

anin

e h

as d

ecid

ed t

o bu

y a

pick

up

tru

ck.H

er c

hoi

ces

incl

ude

eit

her

a V

-6 e

ngi

ne

or a

V-8

en

gin

e,a

stan

dard

cab

or

an e

xten

ded

cab,

and

2-w

hee

l dr

ive

or 4

-wh

eel

driv

e.H

owm

any

poss

ible

mod

els

does

sh

e h

ave

to c

hoo

se f

rom

?8

10.A

mai

l-or

der

com

pan

y th

at s

ells

gar

den

ing

tool

s of

fers

rak

es i

n t

wo

diff

eren

t le

ngt

hs.

Cu

stom

ers

can

als

o ch

oose

eit

her

a w

oode

n,p

last

ic,o

r fi

berg

lass

han

dle

for

the

rake

.H

ow m

any

diff

eren

t ki

nds

of

rake

s ca

n a

cu

stom

er b

uy?

6

11.A

Mex

ican

res

tau

ran

t of

fers

ch

icke

n,b

eef,

or v

eget

aria

n f

ajit

as w

rapp

ed w

ith

eit

her

cor

nor

flo

ur

tort

illa

s,an

d to

pped

wit

h e

ith

er m

ild,

med

ium

,or

hot

sal

sa.H

ow m

any

diff

eren

tch

oice

s of

faj

itas

doe

s a

cust

omer

hav

e?18

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Sta

te w

het

her

th

e ev

ents

are

in

dep

end

ent

or d

epen

den

t.

1.ch

oosi

ng

an i

ce c

ream

fla

vor

and

choo

sin

g a

topp

ing

for

the

ice

crea

min

dep

end

ent

2.ch

oosi

ng

an o

ffen

sive

pla

yer

of t

he

gam

e an

d a

defe

nsi

ve p

laye

r of

th

e ga

me

in a

prof

essi

onal

foo

tbal

l ga

me

ind

epen

den

t

3.F

rom

15

entr

ies

in a

n a

rt c

onte

st,a

cam

p co

un

selo

r ch

oose

s fi

rst,

seco

nd,

and

thir

d pl

ace

win

ner

s.d

epen

den

t

4.Ji

llia

n i

s se

lect

ing

two

mor

e co

urs

es f

or h

er b

lock

sch

edu

le n

ext

sem

este

r.S

he

mu

stse

lect

on

e of

th

ree

mor

nin

g h

isto

ry c

lass

es a

nd

one

of t

wo

afte

rnoo

n m

ath

cla

sses

.in

dep

end

ent

Sol

ve e

ach

pro

ble

m.

5.A

bri

efca

se l

ock

has

3 r

otat

ing

cyli

nde

rs,e

ach

con

tain

ing

10 d

igit

s.H

ow m

any

nu

mer

ical

code

s ar

e po

ssib

le?

1000

6.A

gol

f cl

ub

man

ufa

ctu

rer

mak

es i

ron

s w

ith

7 d

iffe

ren

t sh

aft

len

gth

s,3

diff

eren

t gr

ips,

5di

ffer

ent

lies

,an

d 2

diff

eren

t cl

ub

hea

d m

ater

ials

.How

man

y di

ffer

ent

com

bin

atio

ns

are

offe

red?

210

7.T

her

e ar

e fi

ve d

iffe

ren

t ro

ute

s th

at a

com

mu

ter

can

tak

e fr

om h

er h

ome

to t

he

offi

ce.I

nh

ow m

any

way

s ca

n s

he

mak

e a

rou

nd

trip

if

she

use

s a

diff

eren

t ro

ute

com

ing

than

goin

g?20

8.In

how

man

y w

ays

can

th

e fo

ur

call

let

ters

of

a ra

dio

stat

ion

be

arra

nge

d if

th

e fi

rst

lett

er m

ust

be

W o

r K

an

d n

o le

tter

s re

peat

?27

,600

9.H

ow m

any

7-di

git

phon

e n

um

bers

can

be

form

ed i

f th

e fi

rst

digi

t ca

nn

ot b

e 0

or 1

,an

dan

y di

git

can

be

repe

ated

?8,

000,

000

10.H

ow m

any

7-di

git

phon

e n

um

bers

can

be

form

ed i

f th

e fi

rst

digi

t ca

nn

ot b

e 0,

and

any

digi

t ca

n b

e re

peat

ed?

9,00

0,00

0

11.H

ow m

any

7-di

git

phon

e n

um

bers

can

be

form

ed i

f th

e fi

rst

digi

t ca

nn

ot b

e 0

or 1

,an

d if

no

digi

t ca

n b

e re

peat

ed?

483,

840

12.H

ow m

any

7-di

git

phon

e n

um

bers

can

be

form

ed i

f th

e fi

rst

digi

t ca

nn

ot b

e 0,

and

if n

odi

git

can

be

repe

ated

?54

4,32

0

13.H

ow m

any

6-ch

arac

ter

pass

wor

ds c

an b

e fo

rmed

if

the

firs

t ch

arac

ter

is a

dig

it a

nd

the

rem

ain

ing

5 ch

arac

ters

are

let

ters

th

at c

an b

e re

peat

ed?

118,

813,

760

14.H

ow m

any

6-ch

arac

ter

pass

wor

ds c

an b

e fo

rmed

if

the

firs

t an

d la

st c

har

acte

rs a

redi

gits

an

d th

e re

mai

nin

g ch

arac

ters

are

let

ters

? A

ssu

me

that

an

y ch

arac

ter

can

be

repe

ated

.45

,697

,600

Pra

ctic

e (

Ave

rag

e)

Th

e C

ou

nti

ng

Pri

nci

ple

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1



Readin

g t

o L

earn

Math

em

ati

csT

he

Co

un

tin

g P

rin

cip

le

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1

©G

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3G

lenc

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Lesson 12-1

Pre-

Act

ivit

yH

ow c

an y

ou c

oun

t th

e m

axim

um

nu

mb

er o

f li

cen

se p

late

s a

stat

eca

n i

ssu

e?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-1 a

t th

e to

p of

pag

e 63

2 in

you

r te

xtbo

ok.

Ass

um

e th

at a

ll F

lori

da l

icen

se p

late

s h

ave

thre

e le

tter

s fo

llow

ed b

y th

ree

digi

ts,a

nd

that

th

ere

are

no

rule

s ag

ain

st u

sin

g th

e sa

me

lett

er o

r n

um

ber

mor

e th

an o

nce

.How

man

y ch

oice

s ar

e th

ere

for

each

let

ter?

for

eac

h d

igit

?26

;10

Rea

din

g t

he

Less

on

1.S

ham

im i

s si

gnin

g u

p fo

r h

er c

lass

es.M

ost

of h

er c

lass

es a

re r

equ

ired

,bu

t sh

e h

as t

wo

elec

tive

s.F

or h

er a

rts

clas

s,sh

e ca

n c

hos

e be

twee

n A

rt,B

and,

Ch

oru

s,or

Dra

ma.

For

her

lan

guag

e cl

ass,

she

can

ch

oose

bet

wee

n F

ren

ch,G

erm

an,a

nd

Spa

nis

h.

a.T

o or

gan

ize

her

ch

oice

s,S

ham

im d

ecid

es t

o m

ake

a tr

ee d

iagr

am.L

et A

,B,C

,an

d D

repr

esen

t A

rt,B

and,

Ch

oru

s,an

d D

ram

a,an

d F,

G,a

nd

S r

epre

sen

t F

ren

ch,G

erm

an,

and

Spa

nis

h.C

ompl

ete

the

foll

owin

g di

agra

m.

b.

How

cou

ld S

ham

im h

ave

foun

d th

e nu

mbe

r of

pos

sibl

e co

mbi

nati

ons

wit

hout

mak

ing

atr

ee d

iagr

am?

Sam

ple

an

swer

:Mu

ltip

ly t

he

nu

mb

er o

f ch

oic

es f

or

her

art

scl

ass

by t

he

nu

mb

er o

f ch

oic

es f

or

her

lan

gu

age

clas

s:3

�4

�12

.

2.A

jar

con

tain

s 6

red

mar

bles

,4 b

lue

mar

bles

,an

d 3

yell

ow m

arbl

es.I

ndi

cate

wh

eth

er t

he

even

ts d

escr

ibed

are

dep

end

ent

or i

nd

epen

den

t.

a.A

mar

ble

is d

raw

n o

ut

of t

he

jar

and

is n

ot r

epla

ced.

A s

econ

d m

arbl

e is

dra

wn

.d

epen

den

t

b.

A m

arbl

e is

dra

wn

ou

t of

th

e ja

r an

d is

pu

t ba

ck i

n.T

he

jar

is s

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en.A

sec

ond

mar

ble

is d

raw

n.

ind

epen

den

t

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne

defi

nit

ion

of

ind

epen

den

tis

“n

ot d

eter

min

ed o

r in

flu

ence

d by

som

eon

e or

som

eth

ing

else

.”H

ow c

an t

his

def

init

ion

hel

p yo

u r

emem

ber

the

diff

eren

ce b

etw

een

in

dep

end

ent

and

dep

end

ent

even

ts?

Sam

ple

an

swer

:If

th

e o

utc

om

e o

f o

ne

even

t d

oes

no

taf

fect

or

infl

uen

ce t

he

ou

tco

me

of

ano

ther

,th

e ev

ents

are

ind

epen

den

t.If

the

ou

tco

me

of

on

e ev

ent

do

esaf

fect

or

infl

uen

ce t

he

ou

tco

me

of

ano

ther

,th

e ev

ents

are

dep

end

ent.

FGA

S

AF

AG

AS

FGB

S

BF

BG

BS

FGC

S

CF

CG

CS

FGD

S

DF

DG

DS

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Tree

Dia

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ms

and

th

e P

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er R

ule

If y

ou f

lip

a co

in o

nce

,th

ere

are

two

poss

ible

ou

tcom

es:h

eads

sh

owin

g (H

) or

tai

ls s

how

ing

(T).

Th

e tr

ee d

iagr

am t

o th

e ri

ght

show

s th

e fo

ur

(22 )

poss

ible

ou

tcom

es i

f yo

u f

lip

a co

in t

wic

e.

Flip

2

H T H T

Flip

1

H T

Ou

tco

mes

HH

HT

TH TT

star

t

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1

Dra

w a

tre

e d

iagr

am t

osh

ow a

ll t

he

pos

sib

le o

utc

omes

for

fli

pp

ing

a co

in t

hre

e ti

mes

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t th

e ou

tcom

es.

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ere

are

eigh

t (2

3 ) p

ossi

ble

outc

omes

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hea

ch e

xtra

fli

p,th

e n

um

ber

of o

utc

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do

ubl

es.W

ith

4 f

lips

,th

ere

wou

ld b

e si

xtee

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4 ) o

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.

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2

H T H T

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1

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and

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le.H

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any

pos

sib

le o

utc

omes

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th

ere

ifyo

u d

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on

e m

arb

le a

t ra

nd

om,

rep

lace

it,

and

th

en d

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an

oth

er?

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ere

are

nin

e (3

2 ) p

ossi

ble

outc

omes

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w 2

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e P

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nu

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f an

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erim

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tim

es,a

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if t

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e ar

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poss

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ou

tcom

es e

ach

tim

e,th

ere

are

bnto

tal

poss

ible

ou

tcom

es.

Fin

d t

he

tota

l n

um

ber

of

pos

sib

le o

utc

omes

for

eac

h e

xper

imen

t.U

setr

ee d

iagr

ams

to h

elp

you

.

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ippi

ng

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in 5

tim

es25

2.do

ing

the

mar

ble

expe

rim

ent

6 ti

mes

36

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ippi

ng

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in 8

tim

es28

4.ro

llin

g a

6-si

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62

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llin

g a

6-si

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die

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mes

636.

roll

ing

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side

d di

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mes

438.

roll

ing

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-sid

ed d

ie 2

tim

es12

2


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

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mu

tati

on

s an

d C

om

bin

atio

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NA

ME

____

____

____

____

____

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Wh

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r pe

ople

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arr

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d in

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erta

in o

rder

,th

ear

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ent

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per

mu

tati

on.

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mu

tati

on

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per

mut

atio

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stin

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ts ta

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ith

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hich

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.

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ith

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ects

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ed.

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m a

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ep

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y d

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chos

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ce e

ach

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port

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iffe

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t fo

rmat

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s im

port

ant.

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mu

st f

ind

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nu

mbe

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per

mu

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ken

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tions

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en b

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.

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any

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ts c

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acl

ass

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ince

th

e or

der

of c

hoo

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g th

e st

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is

not

im

port

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you

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st f

ind

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nu

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r of

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bin

atio

ns

of 2

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ude

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tak

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at

a ti

me.

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rmul

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are

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pos

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ow m

any

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s ca

n y

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se 1

vow

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nd

2 c

onso

nan

ts f

rom

ase

t of

26

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er t

iles

? (A

ssu

me

ther

e ar

e 5

vow

els

and

21

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son

ants

.)B

y th

e F

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dam

enta

l C

oun

tin

g P

rin

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e,yo

u c

an m

ult

iply

th

e n

um

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of w

ays

to s

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t on

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wel

an

d th

e n

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of w

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to s

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t 2

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son

ants

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ly t

he

lett

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chos

en m

atte

r,n

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e or

der

in w

hic

h t

hey

wer

e ch

osen

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use

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bin

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ns.

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,1)

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e of

5 v

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s ar

e dr

awn

.C

(21,

2)T

wo

of 2

1 co

nso

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ts a

re d

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,1)

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trac

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0 or

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ify.

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ere

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bin

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2 co

nso

nan

ts.

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luat

e ea

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xpre

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n.

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ME

____

____

____

____

____

____

____

____

____

____

____

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____

____

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____

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Skil

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mu

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ns

NA

ME

____

____

____

____

____

____

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12-2

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120

7.C

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11.C

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12.C

(7,4

)35

Det

erm

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wh

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in

volv

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per

mu

tati

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na

tion

.Th

enfi

nd

th

e n

um

ber

of

pos

sib

ilit

ies.

13.s

eati

ng

8 st

ude

nts

in

8 s

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in

th

e fr

ont

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of

the

sch

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audi

tori

um

per

mu

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;40

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ntr

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e 5

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at

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n;

120

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;56

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;35

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;15

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28

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,a y

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n a

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rp

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n;

24

23.s

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s at

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n;

36

24.a

n a

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5.P

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) �

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,2)

1260

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wh

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er e

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sit

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in

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per

mu

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.Th

enfi

nd

th

e n

um

ber

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pos

sib

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m f

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thle

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126

14.a

n a

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gem

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of t

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lett

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in t

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ap

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120

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gin

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char

ms

on a

bra

cele

t th

at h

as a

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sp,a

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nt,

and

a ba

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uta

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dess

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m 1

0 de

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hat

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17.a

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nu

ally

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40

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m f

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12

sale

speo

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66

19.m

akin

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the

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ith

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ENT

GR

OU

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Hig

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s pl

ann

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acad

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tiva

l.A

ll m

ath

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ses

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l se

nd

2 re

pres

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tive

s to

com

pete

in

th

e m

ath

bow

l.H

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any

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eren

tgr

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of

stu

den

ts c

an b

e ch

osen

fro

m a

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ss o

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stu

den

ts?

120

22.P

HO

TOG

RA

PHY

A p

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is

taki

ng

pict

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s of

a b

ride

an

d gr

oom

an

d th

eir

6at

ten

dan

ts.I

f sh

e ta

kes

phot

ogra

phs

of 3

peo

ple

in a

gro

up,

how

man

y di

ffer

ent

grou

psca

n s

he

phot

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ph?

56

23.A

IRLI

NES

An

air

lin

e is

hir

ing

5 fl

igh

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dan

ts.I

f 8

peop

le a

pply

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th

e jo

b,h

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any

diff

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t gr

oups

of

5 at

ten

dan

ts c

an t

he

airl

ine

hir

e?56

24.S

UB

SCR

IPTI

ON

SA

sch

ool

libr

aria

n w

ould

lik

e to

bu

y su

bscr

ipti

ons

to 7

new

mag

azin

es.H

er b

udg

et,h

owev

er,w

ill

allo

w h

er t

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ly 4

new

su

bscr

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ons.

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man

y di

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ent

grou

ps o

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mag

azin

es c

an s

he

choo

se f

rom

th

e 7

mag

azin

es?

35

Pra

ctic

e (

Ave

rag

e)

Per

mu

tati

on

s an

d C

om

bin

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-2

12-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csP

erm

uta

tio

ns

and

Co

mb

inat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-2

12-2

©G

lenc

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cGra

w-H

ill70

9G

lenc

oe A

lgeb

ra 2

Lesson 12-2

Pre-

Act

ivit

yH

ow d

o p

erm

uta

tion

s an

d c

omb

inat

ion

s ap

ply

to

soft

bal

l?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-2 a

t th

e to

p of

pag

e 63

8 in

you

r te

xtbo

ok.

Su

ppos

e th

at 2

0 st

ude

nts

en

ter

a m

ath

con

test

.In

how

man

y w

ays

can

firs

t,se

con

d,an

d th

ird

plac

es b

e aw

arde

d? (

Wri

te y

our

answ

er a

s a

prod

uct

.D

o n

ot c

alcu

late

th

e pr

odu

ct.)

20 �

19 �

18

Rea

din

g t

he

Less

on

1.In

dica

te w

het

her

eac

h s

itu

atio

n i

nvo

lves

a p

erm

uta

tion

or a

com

bin

atio

n.

a.ch

oosi

ng

five

stu

den

ts f

rom

a c

lass

to

wor

k on

a s

peci

al p

roje

ctco

mb

inat

ion

b.

arra

ngi

ng

five

pic

ture

s in

a r

ow o

n a

wal

lp

erm

uta

tio

n

c.dr

awin

g a

han

d of

13

card

s fr

om a

52-

card

dec

kco

mb

inat

ion

d.

arra

ngi

ng

the

lett

ers

of t

he

wor

d al

gebr

ap

erm

uta

tio

n

2.W

rite

an

exp

ress

ion

th

at c

an b

e u

sed

to c

alcu

late

eac

h o

f th

e fo

llow

ing.

a.n

um

ber

of c

ombi

nat

ion

s of

ndi

stin

ct o

bjec

ts t

aken

rat

a t

ime� (n

�n! r)

!r!

�

b.

nu

mbe

r of

per

mu

tati

ons

of n

obje

cts

of w

hic

h p

are

alik

e an

d q

are

alik

e� pn !q! !

�

c.n

um

ber

of p

erm

uta

tion

s of

ndi

stin

ct o

bjec

ts t

aken

rat

a t

ime� (n

�n! r)

!�

3.F

ive

card

s ar

e dr

awn

fro

m a

sta

nda

rd d

eck

of c

ards

.Su

ppos

e yo

u a

re a

sked

to

dete

rmin

eh

ow m

any

poss

ible

han

ds c

onsi

st o

f on

e h

eart

,tw

o di

amon

ds,a

nd

two

spad

es.

a.W

hic

h o

f th

e fo

llow

ing

wou

ld y

ou u

se t

o so

lve

this

pro

blem

:Fu

nd

amen

tal

Cou

nti

ng

Pri

nci

ple,

perm

uta

tion

s,or

com

bin

atio

ns?

(M

ore

than

on

e of

th

ese

may

app

ly.)

Fu

nd

amen

tal C

ou

nti

ng

Pri

nci

ple

,co

mb

inat

ion

s

b.

Wri

te a

n ex

pres

sion

tha

t in

volv

es t

he n

otat

ion

P(n

,r)

and/

or C

(n,r

) th

at y

ou w

ould

use

to s

olve

th

is p

robl

em.(

Do

not

do

any

calc

ula

tion

s.)

C(1

3,1)

�C

(13,

2) �

C(1

3,2)

Hel

pin

g Y

ou

Rem

emb

er

4.M

any

stu

den

ts h

ave

trou

ble

know

ing

wh

en t

o u

se p

erm

uta

tion

s an

d w

hen

to

use

com

bin

atio

ns

to s

olve

cou

nti

ng

prob

lem

s.H

ow c

an t

he

idea

of

ord

erh

elp

you

to

rem

embe

r th

e di

ffer

ence

bet

wee

n p

erm

uta

tion

s an

d co

mbi

nat

ion

s?

Sam

ple

an

swer

:A

per

mu

tati

on

is a

n a

rran

gem

ent

of

ob

ject

s in

wh

ich

ord

er is

imp

ort

ant.

A c

om

bin

atio

n is

a s

elec

tio

n o

f o

bje

cts

in w

hic

h o

rder

is n

ot

imp

ort

ant.

©G

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w-H

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0G

lenc

oe A

lgeb

ra 2

Co

mb

inat

ion

s an

d P

asca

l’s T

rian

gle

Pas

cal’s

tri

angl

e is

a s

peci

al a

rray

of

nu

mbe

rs i

nve

nte

d by

Bla

ise

Pas

cal

(162

3–16

62).

Th

e va

lues

in

Pas

cal’s

tri

angl

e ca

n b

e fo

un

d u

sin

g th

eco

mbi

nat

ion

s sh

own

bel

ow.

1.E

valu

ate

the

expr

essi

on i

n e

ach

cel

l of

th

e tr

ian

gle.

2.T

he p

atte

rn s

how

s th

e re

lati

onsh

ip b

etw

een

C(n

,r)

and

Pas

cal’s

tri

angl

e.In

gen

eral

,it

is t

rue

that

C(n

,r)

�C

(n,r

�1)

�C

(n�

1,r

�1)

.Com

plet

eth

e pr

oof

of t

his

pro

pert

y.In

eac

h s

tep,

the

den

omin

ator

has

bee

n g

iven

.

C(n

,r)

�C

(n,r

�1)

��

��

��

� � � � �C

(n�

1,r

�1)

(n�

1)!

(r�

1)![

(n�

1) �

(r�

1)]!

(n�

1)!

(r�

1)!(

n�

r)!

n!(

n�

1)(r

�1)

!(n

�r)

!

n!(

r�

1 �

n�

r)(r

�1)

!(n

�r)

!

n!(

n�

r)(r

�1)

!(n

�r)

!n

!(r

�1)

(r�

1)!(

n�

r)!

n!(

n�

r)(r

�1)

!(n

�r

�1)

!(n

�r)

n!(

r�

1)r!

(n�

r)!(

r�

1)

n!

(r�

1)!(

n�

r�

1)!

n!

r!(n

�r)

!

C(1

,0)

1

C(1

,1)

1C

(2,0

)

1

C(2

,1)

2

C(2

,2)

1

C(3

,0)

1

C(3

,1)

3

C(3

,2)

3

C(3

,3)

1C

(4,0

)

1

C(4

,1)

4

C(4

,2)

6

C(4

,3)

4

C(4

,4)

1

C(5

,0)

1

C(5

,1)

5

C(5

,2)

10

C(5

,3)

10

C(5

,4)

5

C(5

,5)

1

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-2

12-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Pro

bab

ility

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3

©G

lenc

oe/M

cGra

w-H

ill71

1G

lenc

oe A

lgeb

ra 2

Lesson 12-3

Pro

bab

ility

an

d O

dd

sIn

pro

babi

lity

,a d

esir

ed o

utc

ome

is c

alle

d a

succ

ess;

any

oth

erou

tcom

e is

cal

led

a fa

ilu

re.

Pro

bab

ility

of

If an

eve

nt c

an s

ucce

ed in

sw

ays

and

fail

in f

way

s, t

hen

the

prob

abili

ties

of s

ucce

ss,

P(S

),

Su

cces

s an

dan

d of

fai

lure

, P

(F),

are

as

follo

ws.

Failu

reP

(S)

�an

d P

(F)

�.

Def

init

ion

If an

eve

nt c

an s

ucce

ed in

sw

ays

and

fail

in f

way

s, t

hen

the

odds

of

succ

ess

and

of f

ailu

re a

re

of

Od

ds

as f

ollo

ws.

Odd

s of

suc

cess

�s

:fO

dds

of f

ailu

re �

f:s

Wh

en 3

coi

ns

are

toss

ed,w

hat

is

the

pro

bab

ilit

y th

at a

t le

ast

2 ar

e h

ead

s?Yo

u c

an u

se a

tre

e di

agra

m t

o fi

nd

the

sam

ple

spac

e.O

f th

e 8

poss

ible

ou

tcom

es,4

hav

e at

lea

st 2

hea

ds.S

o th

e

prob

abil

ity

of t

ossi

ng

at l

east

2 h

eads

is

�4 8�or

�1 2� .

Wh

at i

s th

e p

rob

abil

ity

of p

ick

ing

4 fi

ctio

n b

ook

s an

d 2

bio

grap

hie

sfr

om a

bes

t-se

ller

lis

t th

at c

onsi

sts

of 1

2 fi

ctio

n b

ook

s an

d 6

bio

grap

hie

s?B

y th

e F

un

dam

enta

l C

oun

tin

g P

rin

cipl

e,th

e n

um

ber

of s

ucc

esse

s is

C(1

2,4)

�C

(6,2

).T

he

tota

l n

um

ber

of s

elec

tion

s,s

�f,

of 6

boo

ks i

s C

(18,

6).

P(4

fic

tion

,2 b

iogr

aph

y) �

or a

bou

t 0.

40

Th

e pr

obab

ilit

y of

sel

ecti

ng

4 fi

ctio

n b

ooks

an

d 2

biog

raph

ies

is a

bou

t 40

%.

Fin

d t

he

odd

s of

an

eve

nt

occu

rrin

g,gi

ven

th

e p

rob

abil

ity

of t

he

even

t.

1.�3 7�

3:4

2.�4 5�

4:1

3.� 12 3�

2:11

4.1:

14

Fin

d t

he

pro

bab

ilit

y of

an

eve

nt

occu

rrin

g,gi

ven

th

e od

ds

of t

he

even

t.

5.10

:1�1 10 1�

6.2:

5�2 7�

7.4:

9� 14 3�

8.8:

3� 18 1�

On

e b

ag o

f ca

nd

y co

nta

ins

15 r

ed c

and

ies,

10 y

ello

w c

and

ies,

and

6 g

reen

can

die

s.F

ind

th

e p

rob

abil

ity

of e

ach

sel

ecti

on.

9.pi

ckin

g a

red

can

dy�1 35 1�

10.n

ot p

icki

ng

a ye

llow

can

dy�2 31 1�

11.p

icki

ng

a gr

een

can

dy� 36 1�

12.n

ot p

icki

ng

a re

d ca

ndy

�1 36 1�

1 � 15

C(1

2,4)

�C

(6,2

)�

�C

(18,

6)

HH

HH

HT

HT

HH

TT

TH

HT

HT

TT

HT

TT

H T H T H T H T

H T

H T H T

Firs

tC

oin

Sec

ond

Co

inT

hird

Co

inP

oss

ible

Out

com

es

f� s

�f

s� s

�f

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

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lenc

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w-H

ill71

2G

lenc

oe A

lgeb

ra 2

Pro

bab

ility

Dis

trib

uti

on

sA

ran

dom

var

iab

leis

a v

aria

ble

wh

ose

valu

e is

th

en

um

eric

al o

utc

ome

of a

ran

dom

eve

nt.

A p

rob

abil

ity

dis

trib

uti

onfo

r a

part

icu

lar

ran

dom

vari

able

is

a fu

nct

ion

th

at m

aps

the

sam

ple

spac

e to

th

e pr

obab

ilit

ies

of t

he

outc

omes

in

th

esa

mpl

e sp

ace.

Su

pp

ose

two

dic

e ar

e ro

lled

.Th

e ta

ble

an

d t

he

rela

tive

-fre

qu

ency

his

togr

am s

how

th

e d

istr

ibu

tion

of

the

abso

lute

val

ue

of t

he

dif

fere

nce

of

the

nu

mb

ers

roll

ed.U

se t

he

grap

h t

o d

eter

min

e w

hic

h o

utc

ome

is t

he

mos

t li

kel

y.W

hat

is

its

pro

bab

ilit

y?

Th

e gr

eate

st p

roba

bili

ty i

n t

he

grap

h i

s �1 4� .

Th

e m

ost

like

ly o

utc

ome

is a

dif

fere

nce

of

1 an

d it

s

prob

abil

ity

is �1 4� .

Fou

r co

ins

are

toss

ed.

1.C

ompl

ete

the

tabl

e be

low

to

show

th

e pr

obab

ilit

y di

stri

buti

on o

f th

e n

um

ber

of h

eads

.

2.M

ake

rela

tive

-fre

quen

cy d

istr

ibu

tion

of

the

data

.

10

Hea

ds

Head

s in

Co

in T

oss

23

4

1 4

Probability

3 8 1 8 1 163 165 16Nu

mb

er o

f H

ead

s0

12

34

Pro

bab

ility

� 11 6��1 4�

�3 8��1 4�

� 11 6�

1 4

00

Probability

Dif

fere

nce

Nu

mb

ers

Sh

ow

ing

on

th

e D

ice

12

34

5

1 6 1 12

Dif

fere

nce

01

23

45

Pro

bab

ility

�1 6��1 4�

�1 6��1 6�

�1 6�� 11 2�

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Pro

bab

ility

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Pro

bab

ility

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3

©G

lenc

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cGra

w-H

ill71

3G

lenc

oe A

lgeb

ra 2

Lesson 12-3

Ah

med

is

pos

tin

g 2

ph

otog

rap

hs

on h

is w

ebsi

te.H

e h

as n

arro

wed

his

ch

oice

s to

4la

nd

scap

e p

hot

ogra

ph

s an

d 3

por

trai

ts.I

f h

e ch

oose

s th

e tw

o p

hot

ogra

ph

s at

ran

dom

,fin

d t

he

pro

bab

ilit

y of

eac

h s

elec

tion

.

1.P

(2 p

ortr

ait)

�1 7�2.

P(2

lan

dsca

pe)

�2 7�3.

P(1

of

each

)�4 7�

Th

e C

aru

bas

hav

e a

coll

ecti

on o

f 28

vid

eo m

ovie

s,in

clu

din

g 12

wes

tern

s an

d

16 s

cien

ce f

icti

on.E

lise

sel

ects

3 o

f th

e m

ovie

s at

ran

dom

to

bri

ng

to a

sle

ep-o

ver

at h

er f

rien

d’s

hou

se.F

ind

th

e p

rob

abil

ity

of e

ach

sel

ecti

on.

4.P

(3 w

este

rns)

� 85 15 9�5.

P(3

sci

ence

fic

tion

)� 12 10 7�

6.P

(1 w

este

rn a

nd

2 sc

ien

ce f

icti

on)

�4 90 1�7.

P(2

wes

tern

s an

d 1

scie

nce

fic

tion

)� 28 78 3�

8.P

(3 c

omed

y)0

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(2 s

cien

ce f

icti

on a

nd

2 w

este

rns)

0

For

Exe

rcis

es 1

0–13

,use

th

e ch

art

that

sh

ows

the

clas

s an

d g

end

er s

tati

stic

s fo

r th

e st

ud

ents

tak

ing

an

Alg

ebra

1 o

r A

lgeb

ra 2

cla

ss a

t L

a M

esa

Hig

h S

choo

l.If

a s

tude

nt

taki

ng

Alg

ebra

1 o

r A

lgeb

ra 2

is

sele

cted

at

ran

dom

,fin

d ea

ch p

roba

bili

ty.E

xpre

ss a

s de

cim

als

rou

nde

d to

th

e n

eare

st t

hou

san

dth

.

10.P

(sop

hom

ore/

fem

ale)

0.20

8

11.P

(ju

nio

r/m

ale)

0.14

3

12.P

(fre

shm

an/m

ale)

0.13

6

13.P

(fre

shm

an/f

emal

e)0.

145

Fin

d t

he

odd

s of

an

eve

nt

occu

rrin

g,gi

ven

th

e p

rob

abil

ity

of t

he

even

t.

14.�

5 8�5:

315

.�2 7�

2:5

16.�

3 5�3:

2

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4G

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A b

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ed,a

nd

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ello

w b

alls

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o b

alls

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ind

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rob

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ach

sel

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wo

coin

s ar

ese

lect

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t ra

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ind

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rob

abil

ity

of e

ach

sel

ecti

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rico

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ome

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for

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new

hou

se.T

he

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8 b

ook

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w H

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e fo

r a

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wh

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omly

ch

oose

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d t

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sel

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7–20

,use

th

e ta

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th

at s

how

s th

e ra

nge

of

ver

bal

SA

T s

core

s fo

rfr

esh

men

at

a sm

all

lib

eral

arts

col

lege

.If

a fr

esh

man

stu

den

t is

ch

osen

at

ran

dom

,fin

d e

ach

pro

bab

ilit

y.E

xpre

ss a

s d

ecim

als

rou

nd

ed t

o th

e n

eare

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hou

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.

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nt

occu

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ven

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ds

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28.2

:23

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.2:5

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400–

449

450–

499

500–

549

550–

559

600–

649

650�

Nu

mb

er o

f S

tud

ents

129

275

438

602

620

412

Pra

ctic

e (

Ave

rag

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Pro

bab

ility

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

____

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12-3

12-3



Readin

g t

o L

earn

Math

em

ati

csP

rob

abili

ty

NA

ME

____

____

____

____

____

____

____

____

____

____

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AT

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____

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ER

IOD

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12-3

12-3

©G

lenc

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5G

lenc

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lgeb

ra 2

Lesson 12-3

Pre-

Act

ivit

yW

hat

do

pro

bab

ilit

y an

d o

dd

s te

ll y

ou a

bou

t li

fe’s

ris

ks?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-3 a

t th

e to

p of

pag

e 64

4 in

you

r te

xtbo

ok.

Wh

at i

s th

e pr

obab

ilit

y th

at a

per

son

wil

l n

otbe

str

uck

by

ligh

tnin

g in

agi

ven

yea

r?�7 74 59 0, ,9 09 09 0

�

Rea

din

g t

he

Less

on

1.In

dica

te w

het

her

eac

h o

f th

e fo

llow

ing

stat

emen

ts i

s tr

ue

or f

alse

.

a.If

an

eve

nt

can

nev

er o

ccu

r,it

s pr

obab

ilit

y is

a n

egat

ive

nu

mbe

r.fa

lse

b.

If a

n e

ven

t is

cer

tain

to

hap

pen

,its

pro

babi

lity

is

1.tr

ue

c.If

an

eve

nt

can

su

ccee

d in

sw

ays

and

fail

in

fw

ays,

then

th

e pr

obab

ilit

y of

su

cces

s

is

.fa

lse

d.

If a

n e

ven

t ca

n s

ucc

eed

in s

way

s an

d fa

il i

n f

way

s,th

en t

he

odds

aga

inst

th

e ev

ent

are

s:f.

fals

e

e.A

pro

babi

lity

dis

trib

uti

on i

s a

fun

ctio

n i

n w

hic

h t

he

dom

ain

is

the

sam

ple

spac

e of

an

expe

rim

ent.

tru

e

2.A

wea

ther

for

ecas

t sa

ys t

hat

th

e ch

ance

of

rain

tom

orro

w i

s 40

%.

a.W

rite

th

e pr

obab

ilit

y th

at i

t w

ill

rain

tom

orro

w a

s a

frac

tion

in

low

est

term

s.�2 5�

b.

Wri

te t

he

prob

abil

ity

that

it

wil

l n

ot r

ain

tom

orro

w a

s a

frac

tion

in

low

est

term

s.�3 5�

c.W

hat

are

th

e od

ds i

n f

avor

of

rain

?2:

3

d.

Wh

at a

re t

he

odds

aga

inst

rai

n?

3:2

3.R

efer

to

the

tabl

e in

Exa

mpl

e 4

on p

age

646

in y

our

text

book

.

a.W

hat

oth

er s

um

has

th

e sa

me

prob

abil

ity

as a

su

m o

f 11

?3

b.

Wh

at a

re t

he

odds

of

roll

ing

a su

m o

f 8?

5:31

c.W

hat

are

th

e od

ds a

gain

st r

olli

ng

a su

m o

f 9?

8:1

Hel

pin

g Y

ou

Rem

emb

er

4.A

goo

d w

ay t

o re

mem

ber

som

eth

ing

is t

o ex

plai

n i

t to

som

eon

e el

se.S

upp

ose

that

you

rfr

ien

d R

ober

to i

s h

avin

g tr

oubl

e re

mem

beri

ng

the

diff

eren

ce b

etw

een

pro

babi

lity

an

dod

ds.W

hat

wou

ld y

ou t

ell

him

to

hel

p h

im r

emem

ber

this

eas

ily?

Sam

ple

an

swer

:P

rob

abili

ty g

ives

th

e ra

tio

of

succ

esse

s to

th

e to

tal

nu

mb

er o

f o

utc

om

es,w

hile

od

ds

giv

es t

he

rati

o o

f su

cces

ses

to f

ailu

res.

s � f

©G

lenc

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w-H

ill71

6G

lenc

oe A

lgeb

ra 2

Geo

met

ric

Pro

bab

ility

If a

dar

t,th

row

n a

t ra

ndo

m,h

its

the

tria

ngu

lar

boar

d sh

own

at

the

righ

t,w

hat

is

the

chan

ce t

hat

it

wil

l h

it t

he

shad

ed r

egio

n?

Th

is

chan

ce,a

lso

call

ed a

pro

babi

lity

,can

be

dete

rmin

ed b

y co

mpa

rin

g th

e ar

ea o

f th

e sh

aded

reg

ion

to

the

area

of

the

boar

d.T

his

rat

io

indi

cate

s w

hat

fra

ctio

n o

f th

e to

sses

sh

ould

hit

in

th

e sh

aded

reg

ion

.

� ��1 22 4�

or �1 2�

In g

ener

al,i

f S

is a

su

breg

ion

of

som

e re

gion

R,t

hen

th

e pr

obab

ilit

y,P

(S),

that

a p

oin

t,ch

osen

at

ran

dom

,bel

ongs

to

subr

egio

n S

is g

iven

by

the

foll

owin

g.

P(S

) �

Fin

d t

he

pro

bab

ilit

y th

at a

poi

nt,

chos

en a

t ra

nd

om,b

elon

gs t

o th

esh

aded

su

bre

gion

s of

th

e fo

llow

ing

regi

ons.

1.�1 2�

2.�5 9�

3.�� 4�

Th

e d

art

boa

rd s

how

n a

t th

e ri

ght

has

5 c

once

ntr

ic c

ircl

es

wh

ose

cen

ters

are

als

o th

e ce

nte

r of

th

e sq

uar

e b

oard

.Eac

h

sid

e of

th

e b

oard

is

38 c

m,a

nd

th

e ra

dii

of

the

circ

les

are

2 cm

,5 c

m,8

cm

,11

cm,a

nd

14

cm.A

dar

t h

itti

ng

wit

hin

on

e of

th

e ci

rcu

lar

regi

ons

scor

es t

he

nu

mb

er o

f p

oin

ts i

nd

icat

ed

on t

he

boa

rd,w

hil

e a

hit

an

ywh

ere

else

sco

res

0 p

oin

ts.I

f a

dar

t,th

row

n a

t ra

nd

om,h

its

the

boa

rd,f

ind

th

e p

rob

abil

ity

of s

cori

ng

the

ind

icat

ed n

um

ber

of

poi

nts

.

4.0

poin

ts5.

1 po

int

6.2

poin

ts

�361 3� 61

49�

�� 17 45 4� 4

�� 15 47 4� 4

�

7.3

poin

ts8.

4 po

ints

9.5

poin

ts

� 13 49 4� 4�

� 12 41 4� 4�

� 3� 61�

51

2 34

44

4 4

46

6

64

4

33

5 5

area

of

subr

egio

n S

��

�ar

e of

reg

ion

R

�1 2� (4)

(6)

� �1 2� (8)

(6)

area

of

shad

ed r

egio

n�

��

area

of

tria

ngu

lar

boar

d

44

6

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Mu

ltip

lyin

g P

rob

abili

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4

©G

lenc

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w-H

ill71

7G

lenc

oe A

lgeb

ra 2

Lesson 12-4

Pro

bab

ility

of

Ind

epen

den

t Ev

ents

Pro

bab

ility

of T

wo

If

two

even

ts,

Aan

d B

, ar

e in

depe

nden

t, th

en t

he p

roba

bilit

y of

bot

h oc

curr

ing

isIn

dep

end

ent

Eve

nts

P(A

and

B)

�P

(A)

�P

(B).

In a

boa

rd g

ame

each

pla

yer

has

3

dif

fere

nt-

colo

red

mar

ker

s.T

o m

ove

arou

nd

th

e b

oard

th

e p

laye

r fi

rst

spin

s a

spin

ner

to

det

erm

ine

wh

ich

pie

ce c

an b

e m

oved

.He

or s

he

then

rol

ls a

die

to

det

erm

ine

how

man

y sp

aces

that

col

ored

pie

ce s

hou

ld m

ove.

On

a g

iven

tu

rn w

hat

is

the

pro

bab

ilit

y th

at a

pla

yer

wil

l b

e ab

le t

o m

ove

the

yell

ow p

iece

m

ore

than

2 s

pac

es?

Let

Abe

th

e ev

ent

that

th

e sp

inn

er l

ands

on

yel

low

,an

d le

t B

be t

he

even

t th

at t

he

die

show

s a

nu

mbe

r gr

eate

r th

an 2

.Th

e pr

obab

ilit

y of

Ais

�1 3� ,an

d th

e pr

obab

ilit

y of

Bis

�2 3� .

P(A

and

B)

�P

(A)

�P

(B)

Pro

babi

lity

of in

depe

nden

t ev

ents

��1 3�

��2 3�

or �2 9�

Sub

stitu

te a

nd m

ultip

ly.

Th

e pr

obab

ilit

y th

at t

he

play

er c

an m

ove

the

yell

ow p

iece

mor

e th

an 2

spa

ces

is �2 9� .

A d

ie i

s ro

lled

3 t

imes

.Fin

d t

he

pro

bab

ilit

y of

eac

h e

ven

t.

1.a

1 is

rol

led,

then

a 2

,th

en a

3� 21 16�

2.a

1 or

a 2

is

roll

ed,t

hen

a 3

,th

en a

5 o

r a

6� 51 4�

3.2

odd

nu

mbe

rs a

re r

olle

d,th

en a

6� 21 4�

4.a

nu

mbe

r le

ss t

han

3 i

s ro

lled

,th

en a

3,t

hen

a n

um

ber

grea

ter

than

3� 31 6�

5.A

box

con

tain

s 5

tria

ngl

es,6

cir

cles

,an

d 4

squ

ares

.If

a fi

gure

is

rem

oved

,rep

lace

d,an

da

seco

nd

figu

re i

s pi

cked

,wh

at i

s th

e pr

obab

ilit

y th

at a

tri

angl

e an

d th

en a

cir

cle

wil

l be

pic

ked?

� 12 5�o

r ab

ou

t 0.

13

6.A

bag

con

tain

s 5

red

mar

bles

an

d 4

wh

ite

mar

bles

.A m

arbl

e is

sel

ecte

d fr

om t

he

bag,

then

rep

lace

d,an

d a

seco

nd

sele

ctio

n i

s m

ade.

Wh

at i

s th

e pr

obab

ilit

y of

sel

ecti

ng

2 re

dm

arbl

es?

�2 85 1�o

r ab

ou

t 0.

31

7.A

jar

con

tain

s 7

lem

on ja

wbr

eake

rs,3

ch

erry

jaw

brea

kers

,an

d 8

rain

bow

jaw

brea

kers

.W

hat

is

the

prob

abil

ity

of s

elec

tin

g 2

lem

on ja

wbr

eake

rs i

n s

ucc

essi

on p

rovi

din

g th

eja

wbr

eake

r dr

awn

fir

st i

s th

en r

epla

ced

befo

re t

he

seco

nd

is d

raw

n?

� 34 29 4�o

r ab

ou

t0.1

5

blue

red

yello

w

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill71

8G

lenc

oe A

lgeb

ra 2

Pro

bab

ility

of

Dep

end

ent

Even

ts

Pro

bab

ility

of T

wo

If

two

even

ts,

Aan

d B

, ar

e de

pend

ent,

then

the

pro

babi

lity

of b

oth

even

ts o

ccur

ring

isD

epen

den

t E

ven

tsP

(Aan

d B

) �

P(A

) �

P(B

follo

win

g A

).

Th

ere

are

7 d

imes

an

d 9

pen

nie

s in

a w

alle

t.S

up

pos

e tw

o co

ins

are

to b

e se

lect

ed a

t ra

nd

om,w

ith

out

rep

laci

ng

the

firs

t on

e.F

ind

th

e p

rob

abil

ity

ofp

ick

ing

a p

enn

y an

d t

hen

a d

ime.

Bec

ause

th

e co

in i

s n

ot r

epla

ced,

the

even

ts a

re d

epen

den

t.

Th

us,

P(A

and

B)

�P

(A)

�P

(Bfo

llow

ing

A).

P(p

enn

y,th

en d

ime)

�P

(pen

ny)

�P

(dim

e fo

llow

ing

pen

ny)

� 19 6��

� 17 5��

�2 81 0�

Th

e pr

obab

ilit

y is

�2 81 0�or

abo

ut

0.26

Wh

at i

s th

e p

rob

abil

ity

of d

raw

ing,

wit

hou

t re

pla

cem

ent,

3 h

eart

s,th

en a

sp

ade

from

a s

tan

dar

d d

eck

of

card

s?S

ince

th

e ca

rds

are

not

rep

lace

d,th

e ev

ents

are

dep

ende

nt.

Let

H r

epre

sen

t a

hea

rt a

nd

Sre

pres

ent

a sp

ade.

P(H

,H,H

,S)

�P

(H)

�P

(H f

ollo

win

g H

) �

P(H

fol

low

ing

2 H

s) �

P(S

fol

low

ing

3 H

s)

��1 53 2�

��1 52 1�

��1 51 0�

��1 43 9�

or a

bou

t 0.

003

Th

e pr

obab

ilit

y is

abo

ut

0.00

3 of

dra

win

g 3

hea

rts,

then

a s

pade

.

Fin

d e

ach

pro

bab

ilit

y.

1.T

he

cup

on S

oph

ie’s

des

k h

olds

4 r

ed p

ens

and

7 bl

ack

pen

s.W

hat

is

the

prob

abil

ity

ofh

er s

elec

tin

g fi

rst

a bl

ack

pen

,th

en a

red

on

e?�1 54 5�

or

abo

ut

0.25

2.W

hat

is

the

prob

abil

ity

of d

raw

ing

two

card

s sh

owin

g od

d n

um

bers

fro

m a

set

of

card

sth

at s

how

th

e fi

rst

20 c

oun

tin

g n

um

bers

if

the

firs

t ca

rd i

s n

ot r

epla

ced

befo

re t

he

seco

nd

is c

hos

en?

� 39 8�o

r ab

ou

t 0.

24

3.T

her

e ar

e 3

quar

ters

,4 d

imes

,an

d 7

nic

kels

in

a c

han

ge p

urs

e.S

upp

ose

3 co

ins

are

sele

cted

wit

hou

t re

plac

emen

t.W

hat

is

the

prob

abil

ity

of s

elec

tin

g a

quar

ter,

then

a d

ime,

and

then

a n

icke

l?� 21 6�

or

abo

ut

0.04

4.A

bas

ket

con

tain

s 4

plu

ms,

6 pe

ach

es,a

nd

5 or

ange

s.W

hat

is

the

prob

abil

ity

of p

icki

ng

2 or

ange

s,th

en a

pea

ch i

f 3

piec

es o

f fr

uit

are

sel

ecte

d at

ran

dom

?� 94 1�

or

abo

ut

0.04

5.A

ph

otog

raph

er h

as t

aken

8 b

lack

an

d w

hit

e ph

otog

raph

s an

d 10

col

or p

hot

ogra

phs

for

abr

och

ure

.If

4 ph

otog

raph

s ar

e se

lect

ed a

t ra

ndo

m,w

hat

is

the

prob

abil

ity

of p

icki

ng

firs

t2

blac

k an

d w

hit

e ph

otog

raph

s,th

en 2

col

or p

hot

ogra

phs?

� 17 02�o

r ab

ou

t 0.

07

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Mu

ltip

lyin

g P

rob

abili

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Mu

ltip

lyin

g P

rob

abili

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4

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Lesson 12-4

A d

ie i

s ro

lled

tw

ice.

Fin

d e

ach

pro

bab

ilit

y.

1.P

(5,t

hen

6)

� 31 6�2.

P(n

o 2s

)�2 35 6�

3.P

(tw

o 1s

)� 31 6�

4.P

(an

y n

um

ber,

then

not

5)

�5 6�5.

P(4

,th

en n

ot 6

)� 35 6�

6.P

(not

1,t

hen

not

2)

�2 35 6�

A b

oard

gam

e u

ses

a se

t of

6 d

iffe

ren

t ca

rds.

Eac

h c

ard

dis

pla

ys o

ne

of t

he

foll

owin

gfi

gure

s:a

star

,a s

qu

are,

a ci

rcle

,a d

iam

ond

,a r

ecta

ngl

e,or

a p

enta

gon

.Th

e ca

rds

are

pla

ced

fac

e d

own

,an

d a

pla

yer

choo

ses

two

card

s.F

ind

eac

h p

rob

abil

ity.

7.P

(cir

cle,

then

sta

r),i

f n

o re

plac

emen

t oc

curs

� 31 0�

8.P

(dia

mon

d,th

en s

quar

e),i

f re

plac

emen

t oc

curs

� 31 6�

9.P

(2 p

olyg

ons)

,if

repl

acem

ent

occu

rs�2 35 6�

10.P

(2 p

olyg

ons)

,if

no

repl

acem

ent

occu

rs�2 3�

11.P

(cir

cle,

then

hex

agon

),if

no

repl

acem

ent

occu

rs0

Det

erm

ine

wh

eth

er t

he

even

ts a

re i

nd

epen

den

tor

dep

end

ent.

Th

en f

ind

eac

hp

rob

abil

ity.

12.A

mix

ed b

ox o

f h

erba

l te

abag

s co

nta

ins

2 le

mon

tea

bags

,3 o

ran

ge-m

ango

tea

bags

,3

cham

omil

e te

abag

s,an

d 1

apri

cot-

gin

ger

teab

ag.K

evin

ch

oose

s 2

teab

ags

at r

ando

m t

obr

ing

to w

ork

wit

h h

im.W

hat

is

the

prob

abil

ity

that

he

firs

t ch

oose

s a

lem

on t

eaba

g an

dth

en a

ch

amom

ile

teab

ag?

dep

end

ent;

� 11 2�

13.T

he

char

t sh

ows

the

sele

ctio

n o

f ol

ive

oils

th

at

Has

ha

fin

ds i

n a

spe

cial

ty f

oods

cat

alog

.If

she

ran

dom

ly s

elec

ts o

ne

type

of

oil,

then

ran

dom

lyse

lect

s an

oth

er,d

iffe

ren

t oi

l,w

hat

is

the

prob

abil

ity

that

bot

h s

elec

tion

s ar

e do

mes

tic,

firs

t co

ld p

ress

ed o

ils?

dep

end

ent;

� 82 21 0�

For

Exe

rcis

es 1

4 an

d 1

5,tw

o th

ird

s of

th

e ar

ea o

f th

e sp

inn

er

earn

s yo

u 5

0 p

oin

ts.S

up

pos

e yo

u s

pin

th

e sp

inn

er t

wic

e.

14.S

ketc

h a

tre

e di

agra

m s

how

ing

all

of t

he

poss

ibil

itie

s.U

se i

t to

fin

d th

e pr

obab

ilit

y of

spin

nin

g 50

poi

nts

,th

en 1

00 p

oin

ts.

�2 9�

15.W

hat

is

the

prob

abil

ity

that

you

get

100

poi

nts

on

eac

h s

pin

?�1 9�

50 100

50 100

2 3

50 100

2 3 2 31 3

1 31 3

100

50

Typ

e o

f O

ilD

om

esti

cIm

po

rted

Pur

e2

5

Col

d P

ress

ed4

8

Firs

t C

old

Pre

ssed

715

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lenc

oe A

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A d

ie i

s ro

lled

th

ree

tim

es.F

ind

eac

h p

rob

abil

ity.

1.P

(th

ree

4s)

� 21 16�2.

P(n

o 4s

)�1 22 15 6�

3.P

(2,t

hen

3,t

hen

1)

� 21 16�4.

P(t

hre

e di

ffer

ent

even

nu

mbe

rs)

� 31 6�

5.P

(an

y n

um

ber,

then

5,t

hen

5)

� 31 6�6.

P(e

ven

nu

mbe

r,th

en o

dd n

um

ber,

then

1)

� 21 4�

Th

ere

are

3 n

ick

els,

2 d

imes

,an

d 5

qu

arte

rs i

n a

pu

rse.

Th

ree

coin

s ar

e se

lect

ed i

nsu

cces

sion

at

ran

dom

.Fin

d t

he

pro

bab

ilit

y.

7.P

(nic

kel,

then

dim

e,th

en q

uar

ter)

,if

no

repl

acem

ent

occu

rs� 21 4�

8.P

(nic

kel,

then

dim

e,th

en q

uar

ter)

,if

repl

acem

ent

occu

rs� 13 00�

9.P

(2 n

icke

ls,t

hen

1 q

uar

ter)

,if

no

repl

acem

ent

occu

rs� 21 4�

10.P

(3 d

imes

),if

rep

lace

men

t oc

curs

� 11 25�

11.P

(3 d

imes

),if

no

repl

acem

ent

occu

rs0

For

Exe

rcis

es 1

2 an

d 1

3,d

eter

min

e w

het

her

th

e ev

ents

are

in

dep

end

ent

ord

epen

den

t.T

hen

fin

d e

ach

pro

bab

ilit

y.

12.S

eren

a is

cre

atin

g a

pain

ting

.She

wan

ts t

o us

e 2

mor

e co

lors

.She

cho

oses

ran

dom

ly f

rom

6 sh

ades

of

red,

10 s

had

es o

f gr

een

,4 s

had

es o

f ye

llow

,4 s

had

es o

f pu

rple

,an

d 6

shad

esof

blu

e.W

hat

is

the

prob

abil

ity

that

sh

e ch

oose

s 2

shad

es o

f gr

een

?d

epen

den

t;� 23 9�

13.K

ersh

el’s

mot

her

is

shop

pin

g at

a b

aker

y.T

he

own

er o

ffer

s K

ersh

el a

coo

kie

from

a ja

rco

nta

inin

g 22

ch

ocol

ate

chip

coo

kies

,18

suga

r co

okie

s,an

d 15

oat

mea

l co

okie

s.W

ith

out

look

ing,

Ker

shel

sel

ects

on

e,dr

ops

it b

ack

in,a

nd

then

ran

dom

ly s

elec

ts a

not

her

.Wh

at i

sth

e pr

obab

ilit

y th

at n

eith

er s

elec

tion

was

a c

hoc

olat

e ch

ip c

ooki

e?in

dep

end

ent;

� 29 5�

14.M

ETEO

RO

LOG

YT

he

Fade

eva’

s ar

e pl

ann

ing

a 3-

day

vaca

tion

to

the

mou

nta

ins.

Alo

ng-

ran

ge f

orec

ast

repo

rts

that

th

e pr

obab

ilit

y of

rai

n e

ach

day

is

10%

.Ass

um

ing

that

the

dail

y pr

obab

ilit

ies

of r

ain

are

in

depe

nde

nt,

wh

at i

s th

e pr

obab

ilit

y th

at t

her

e is

no

rain

on

th

e fi

rst

two

days

,bu

t th

at i

t ra

ins

on t

he

thir

d da

y?� 18 01 00�

RA

ND

OM

NU

MB

ERS

For

Exe

rcis

es 1

5 an

d 1

6,u

se t

he

foll

owin

g in

form

atio

n.

An

ita

has

a l

ist

of 2

0 jo

bs a

rou

nd

the

hou

se t

o do

,an

d pl

ans

to d

o 3

of t

hem

tod

ay.S

he

assi

gns

each

job

a n

um

ber

from

1 t

o 20

,an

dse

ts h

er c

alcu

lato

r to

gen

erat

e ra

ndo

m n

um

bers

fro

m 1

to

20,w

hic

hca

n r

eocc

ur.

Of

the

jobs

,3 a

re o

uts

ide,

and

the

rest

are

in

side

.

15.S

ketc

h a

tre

e di

agra

m s

how

ing

all

of t

he

poss

ibil

itie

s th

at t

he

firs

t th

ree

nu

mbe

rs g

ener

ated

cor

resp

ond

to

insi

de jo

bs o

r ou

tsid

e jo

bs.U

se i

t to

fin

d th

e pr

obab

ilit

y th

at t

he

firs

t tw

o n

um

bers

cor

resp

ond

to i

nsi

de jo

bs,

and

the

thir

d to

an

ou

tsid

e jo

b.0.

1083

75

16.W

hat

is

the

prob

abil

ity

that

th

e n

um

ber

gen

erat

ed

corr

espo

nds

to

an o

uts

ide

job

thre

e ti

mes

in

a r

ow?

0.00

3375

I O

0.85

0.15

I O

0.85

0.15

I O

0.85

0.15

I O

0.85

0.15

I O

0.85

0.15

I O

0.85

0.15

I O

0.85

0.15

Pra

ctic

e (

Ave

rag

e)

Mu

ltip

lyin

g P

rob

abili

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csM

ult

iply

ing

Pro

bab

iliti

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4

©G

lenc

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w-H

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lenc

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Lesson 12-4

Pre-

Act

ivit

yH

ow d

oes

pro

bab

ilit

y ap

ply

to

bas

ket

bal

l?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-4 a

t th

e to

p of

pag

e 65

1 in

you

r te

xtbo

ok.

Wri

te t

he

prob

abil

ity

that

Reg

gie

Mil

ler

mad

e a

free

-th

row

sh

ot d

uri

ng

the

1998

�99

sea

son

as

a fr

acti

on i

n l

owes

t te

rms.

(You

r an

swer

sh

ould

not

incl

ude

a d

ecim

al.)

�1 28 03 0�

Rea

din

g t

he

Less

on

1.A

bag

con

tain

s 4

yell

ow b

alls

,5 r

ed b

alls

,1 w

hit

e ba

ll,a

nd

2 bl

ack

ball

s.A

ball

is

draw

nfr

om t

he

bag

and

is n

ot r

epla

ced.

Ase

con

d ba

ll i

s dr

awn

.

a.L

et Y

be t

he

even

t “f

irst

bal

l is

yel

low

”an

d B

be t

he

even

t “s

econ

d ba

ll i

s bl

ack.

”A

reth

ese

even

ts i

nd

epen

den

tor

dep

end

ent?

dep

end

ent

b.

Tel

l w

hic

h f

orm

ula

you

wou

ld u

se t

o fi

nd

the

prob

abil

ity

that

th

e fi

rst

ball

is

yell

owan

d th

e se

con

d ba

ll i

s bl

ack.

C

A.

P(Y

and

B)

�

B.P

(Yan

d B

) �

P(Y

) �

P(B

)

C.

P(Y

and

B)

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(Y)

�P

(Bfo

llow

ing

Y)

c.W

hic

h e

quat

ion

sh

ows

the

corr

ect

calc

ula

tion

of

this

pro

babi

lity

?B

A.

�1 3��

� 12 1��

�1 37 3�B

.�1 3�

�� 12 1�

�� 32 3�

C.

�1 3��

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.�1 3�

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d.

Wh

ich

equ

atio

n s

how

s th

e co

rrec

t ca

lcu

lati

on o

f th

e pr

obab

ilit

y th

at i

f th

ree

ball

s ar

edr

awn

in

su

cces

sion

wit

hou

t re

plac

emen

t,al

l th

ree

wil

l be

red

?B

A.

� 15 2��

� 15 2��

� 15 2��

� 11 72 25 8�

B.�

15 2��

� 14 1��

� 13 0��

� 21 2�

C.

� 15 2��

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Hel

pin

g Y

ou

Rem

emb

er

2.S

ome

stu

den

ts h

ave

trou

ble

rem

embe

rin

g a

lot

of f

orm

ula

s,so

th

ey t

ry t

o ke

ep t

he

nu

mbe

r of

for

mu

las

they

hav

e to

kn

ow t

o a

min

imu

m.C

an y

ou l

earn

just

on

e fo

rmu

lath

at w

ill

allo

w y

ou t

o fi

nd

prob

abil

itie

s fo

r bo

th i

nde

pen

den

t an

d de

pen

den

t ev

ents

?E

xpla

in y

our

reas

onin

g.S

amp

le a

nsw

er:

Just

rem

emb

er t

he

form

ula

fo

rd

epen

den

t ev

ents

:P

(Aan

d B

) �

P(A

) �

P(B

follo

win

g A

).W

hen

th

eev

ents

are

ind

epen

den

t,P

(Bfo

llow

ing

A)

�P

(B),

so t

he

form

ula

fo

rd

epen

den

t ev

ents

sim

plif

ies

to P

(Aan

d B

) �

P(A

) �

P(B

),w

hic

h is

th

eco

rrec

t fo

rmu

la f

or

ind

epen

den

t ev

ents

.

P(Y

)�

�P

(Y)

�P

(B)

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Co

nd

itio

nal

Pro

bab

ility

Su

ppos

e a

pair

of

dice

is

thro

wn

.It

is k

now

n t

hat

th

e su

m i

s gr

eate

r th

anse

ven

.Fin

d th

e pr

obab

ilit

y th

at t

he

dice

mat

ch.

Th

e pr

obab

ilit

y of

an

eve

nt

give

n t

he

occu

rren

ce o

f an

oth

er e

ven

t is

cal

led

con

dit

ion

al p

roba

bili

ty.T

he

con

diti

onal

pro

babi

lity

of

even

t A

,th

e di

cem

atch

,giv

en e

ven

t B

,th

eir

sum

is

grea

ter

than

sev

en,i

s de

not

ed P

(A/B

).

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4

Th

ere

are

15 s

um

s gr

eate

r th

an s

even

an

dth

ere

are

36 p

ossi

ble

pair

s al

toge

ther

.

P(B

) �

�1 35 6�

Th

ere

are

thre

e m

atch

ing

pair

s gr

eate

rth

an s

even

.

P(A

and

B)

�� 33 6�

P(A

/B)

�

P(A

/B)

�or

�1 5�

Th

e co

ndi

tion

al p

roba

bili

ty i

s �1 5� .

A c

ard

is

dra

wn

fro

m a

sta

nd

ard

dec

k o

f 52

an

d i

s fo

un

d t

o b

e re

d.

Giv

en t

hat

eve

nt,

fin

d e

ach

of

the

foll

owin

g p

rob

abil

itie

s.

1.P

(hea

rt)

�1 2�2.

P(a

ce)

� 11 3�3.

P(f

ace

card

)� 13 3�

4.P

(jac

k or

ten

)� 12 3�

5.P

(six

of

spad

es)

06.

P(s

ix o

f h

eart

s)� 21 6�

A s

por

ts s

urv

ey t

aken

at

Sti

rers

Hig

h S

choo

l sh

ows

that

48%

of

the

resp

ond

ents

lik

ed s

occe

r,66

% l

iked

bas

ket

bal

l,an

d 3

8% l

iked

hoc

key

.A

lso,

30%

lik

ed s

occe

r an

d b

ask

etb

all,

22%

lik

ed b

ask

etb

all

and

hoc

key

and

28%

lik

ed s

occe

r an

d h

ock

ey.F

inal

ly,1

2% l

iked

all

th

ree

spor

ts.

Fin

d e

ach

of

the

foll

owin

g p

rob

abil

itie

s.

7.T

he

prob

abil

ity

Meg

lik

es s

occe

r if

sh

e li

kes

bask

etba

ll.

�3 60 6�o

r � 15 1�

8.T

he

prob

abil

ity

Bif

f li

kes

bask

etba

ll i

f h

e li

kes

socc

er.

�3 40 8�o

r �5 8�

9.T

he

prob

abil

ity

Mu

ffy

like

s h

ocke

y if

sh

e li

kes

bask

etba

ll.

�2 62 6�o

r �1 3�

10.T

he

prob

abil

ity

Gre

g li

kes

hoc

key

and

bask

etba

ll i

f h

e li

kes

socc

er.

�1 42 8�o

r �1 4�

� 33 6�

� �1 35 6�P(A

and

B)

��

P(B

)



Stu

dy G

uid

e a

nd I

nte

rven

tion

Ad

din

g P

rob

abili

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-5

12-5

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Lesson 12-5

Mu

tual

ly E

xclu

sive

Eve

nts

Eve

nts

th

at c

ann

ot o

ccu

r at

th

e sa

me

tim

e ar

e ca

lled

mu

tual

ly e

xclu

sive

eve

nts

.

Pro

bab

ility

of

Mu

tual

ly

If tw

o ev

ents

, A

and

B,

are

mut

ually

exc

lusi

ve,

then

Exc

lusi

ve E

ven

tsP

(Aor

B)

�P

(A)

�P

(B).

Th

is f

orm

ula

can

be

exte

nde

d to

an

y n

um

ber

of m

utu

ally

exc

lusi

ve e

ven

ts.

To

choo

se a

n a

fter

noo

n a

ctiv

ity,

sum

mer

cam

per

s p

ull

sli

ps

ofp

aper

ou

t of

a h

at.T

oday

th

ere

are

25 s

lip

s fo

r a

nat

ure

wal

k,3

5 sl

ips

for

swim

min

g,an

d 3

0 sl

ips

for

arts

an

d c

raft

s.W

hat

is

the

pro

bab

ilit

y th

at a

cam

per

wil

l p

ull

a s

lip

for

a n

atu

re w

alk

or

for

swim

min

g?T

hes

e ar

e m

utu

ally

exc

lusi

ve e

ven

ts.N

ote

that

th

ere

is a

tot

al o

f 90

sli

ps.

P(n

atu

re w

alk

or s

wim

min

g) �

P(n

atu

re w

alk)

�P

(sw

imm

ing)

��2 95 0�

��3 95 0�

or �2 3�

Th

e pr

obab

ilit

y of

a c

ampe

r’s

pull

ing

out

a sl

ip f

or a

nat

ure

wal

k or

for

sw

imm

ing

is �2 3� .

By

the

tim

e on

e te

nt

of 6

cam

per

s ge

ts t

o th

e fr

ont

of t

he

lin

e,th

ere

are

only

10

nat

ure

wal

k s

lip

s an

d 1

5 sw

imm

ing

slip

s le

ft.W

hat

is

the

pro

bab

ilit

yth

at m

ore

than

4 o

f th

e 6

cam

per

s w

ill

choo

se a

sw

imm

ing

slip

?

P(m

ore

than

4 s

wim

mer

s) �

P(5

sw

imm

ers)

�P

(6 s

wim

mer

s)

��

�0.

2T

he

prob

abil

ity

of m

ore

than

4 o

f th

e ca

mpe

rs s

wim

min

g is

abo

ut

0.2.

Fin

d e

ach

pro

bab

ilit

y.

1.A

bag

con

tain

s 45

dye

d eg

gs:1

5 ye

llow

,12

gree

n,a

nd

18 r

ed.W

hat

is

the

prob

abil

ity

ofse

lect

ing

a gr

een

or

a re

d eg

g?�2 3�

2.T

he

lett

ers

from

th

e w

ords

LO

VE

an

d L

IVE

are

pla

ced

on c

ards

an

d pu

t in

a b

ox.W

hat

is t

he

prob

abil

ity

of s

elec

tin

g an

L o

r an

O f

rom

th

e bo

x?�3 8�

3.A

pai

r of

dic

e is

rol

led,

and

the

two

nu

mbe

rs a

re a

dded

.Wh

at i

s th

e pr

obab

ilit

y th

at t

he

sum

is

eith

er a

5 o

r a

7?� 15 8�

or

abo

ut

0.28

4.A

bow

l h

as 1

0 w

hol

e w

hea

t cr

acke

rs,1

6 se

sam

e cr

acke

rs,a

nd

14 r

ye c

risp

s.If

a p

erso

npi

cks

a cr

acke

r at

ran

dom

,wh

at i

s th

e pr

obab

ilit

y of

pic

kin

g ei

ther

a s

esam

e cr

acke

r or

a ry

e cr

isp?

�3 4�

5.A

n ar

t bo

x co

ntai

ns 1

2 co

lore

d pe

ncil

s an

d 20

pas

tels

.If

5 dr

awin

g im

plem

ents

are

cho

sen

at r

ando

m,w

hat

is

the

prob

abil

ity

that

at

leas

t 4

of t

hem

are

pas

tels

?ab

ou

t 0.

37

C(1

0,0)

�C

(15,

6)�

��

C(2

5,6)

C(1

0,1)

�C

(15,

5)�

��

C(2

5,6)

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

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Incl

usi

ve E

ven

ts

Pro

bab

ility

of

Incl

usi

ve E

ven

tsIf

two

even

ts,

Aan

d B

, ar

e in

clus

ive,

P(A

or B

) �

P(A

) �

P(B

) �

P(A

and

B).

Wh

at i

s th

e p

rob

abil

ity

of d

raw

ing

a fa

ce c

ard

or

a b

lack

car

d f

rom

a st

and

ard

dec

k o

f ca

rds?

Th

e tw

o ev

ents

are

in

clu

sive

,sin

ce a

car

d ca

n b

e bo

th a

fac

e ca

rd a

nd

a bl

ack

card

.

P(f

ace

card

or

blac

k ca

rd)

�P

(fac

e ca

rd)

�P

(bla

ck c

ard)

�P

(bla

ck f

ace

card

)

�� 13 3�

��1 2�

�� 23 6�

�� 18 3�

or a

bou

t 0.

62

Th

e pr

obab

ilit

y of

dra

win

g ei

ther

a f

ace

card

or

a bl

ack

card

is

abou

t 0.

62

Fin

d e

ach

pro

bab

ilit

y.

1.W

hat

is

the

prob

abil

ity

of d

raw

ing

a re

d ca

rd o

r an

ace

fro

m a

sta

nda

rd d

eck

of c

ards

?

� 17 3�o

r ab

ou

t 0.

54

2.T

hre

e ca

rds

are

sele

cted

fro

m a

sta

nda

rd d

eck

of 5

2 ca

rds.

Wh

at i

s th

e pr

obab

ilit

y of

sele

ctin

g a

kin

g,a

quee

n,o

r a

red

card

?

�1 25 6�o

r ab

ou

t 0.

58

3.T

he

lett

ers

of t

he

alph

abet

are

pla

ced

in a

bag

.Wh

at i

s th

e pr

obab

ilit

y of

sel

ecti

ng

avo

wel

or

one

of t

he

lett

ers

from

th

e w

ord

QU

IZ?

� 27 6�o

r ab

ou

t 0.

27

4.A

pai

r of

dic

e is

rol

led.

Wh

at i

s th

e pr

obab

ilit

y th

at t

he

sum

is

odd

or a

mu

ltip

le o

f 3?

� 17 1�o

r ab

ou

t 0.

64

5.T

he

Ven

n d

iagr

am a

t th

e ri

ght

show

s th

e n

um

ber

of

ju

nio

rs o

n v

arsi

ty s

port

s te

ams

at E

lmw

ood

Hig

h S

choo

l.S

ome

ath

lete

s ar

e on

var

sity

tea

ms

for

one

seas

on o

nly

,so

me

ath

lete

s fo

r tw

o se

ason

s,an

d so

me

for

all

thre

ese

ason

s.If

a v

arsi

ty a

thle

te i

s ch

osen

at

ran

dom

fro

m

the

jun

ior

clas

s,w

hat

is

the

prob

abil

ity

that

he

or s

he

play

s a

fall

or

win

ter

spor

t?�1 13 6�

Win

ter

Jun

iors

Pla

yin

g V

arsi

ty S

po

rts

Sp

rin

g

Fall

5

6

83 5

41

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Ad

din

g P

rob

abili

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-5

12-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Ad

din

g P

rob

abili

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-5

12-5

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Lesson 12-5

Eli

has

10

bas

ebal

l ca

rds

of 1

0 d

iffe

ren

t p

laye

rs i

n h

is p

ock

et.T

hre

e p

laye

rs a

rep

itch

ers,

5 ar

e ou

tfie

lder

s,an

d 2

are

cat

cher

s.If

Eli

ran

dom

ly s

elec

ts a

car

d t

otr

ade,

fin

d e

ach

pro

bab

ilit

y.

1.P

(pit

cher

or

outf

ield

er)

�4 5�2.

P(p

itch

er o

r ca

tch

er)

�1 2�3.

P(o

utf

ield

er o

r ca

tch

er)

� 17 0�

A d

ie i

s ro

lled

.Fin

d e

ach

pro

bab

ilit

y.

4.P

(5 o

r 6)

�1 3�5.

P(a

t le

ast

a 3)

�2 3�6.

P(l

ess

than

4)

�1 2�

Det

erm

ine

wh

eth

er t

he

even

ts a

re m

utu

all

y ex

clu

sive

or i

ncl

usi

ve.T

hen

fin

d t

he

pro

bab

ilit

y.

7.A

die

is

roll

ed.W

hat

is

the

prob

abil

ity

of r

olli

ng

a 3

or a

4?

mu

tual

ly e

xclu

sive

;�1 3�

8.A

die

is

roll

ed.W

hat

is

the

prob

abil

ity

of r

olli

ng

an e

ven

nu

mbe

r or

a 4

?in

clu

sive

;�1 2�

9.A

car

d is

dra

wn

fro

m a

sta

nda

rd d

eck

of c

ards

.Wh

at i

s th

e pr

obab

ilit

y of

dra

win

g a

kin

gor

a q

uee

n?

mu

tual

ly e

xclu

sive

;� 12 3�

10.A

car

d is

dra

wn

fro

m a

sta

nda

rd d

eck

of c

ards

.Wh

at i

s th

e pr

obab

ilit

y of

dra

win

g a

jack

or a

hea

rt?

incl

usi

ve;

� 14 3�

11.T

he

soph

omor

e cl

ass

is s

elli

ng

Mot

her

’s D

ay p

lan

ts t

o ra

ise

mon

ey.S

usa

n’s

pri

ze f

orbe

ing

the

top

sell

er o

f pl

ants

is

a ch

oice

of

a bo

ok,a

CD

,or

a vi

deo.

Sh

e ca

n c

hoo

se f

rom

6 bo

oks,

3 C

Ds,

and

5 vi

deos

.Wh

at i

s th

e pr

obab

ilit

y th

at S

usa

n s

elec

ts a

boo

k or

a C

D?

mu

tual

ly e

xclu

sive

;� 19 4�

A s

pin

ner

nu

mb

ered

1�

10 i

s sp

un

.Fin

d e

ach

pro

bab

ilit

y.

12.P

(les

s th

an 5

or

even

)� 17 0�

13.P

(eve

n o

r od

d)1

14.P

(pri

me

or e

ven

)�4 5�

Tw

o ca

rds

are

dra

wn

fro

m a

sta

nd

ard

dec

k o

f ca

rds.

Fin

d e

ach

pro

bab

ilit

y.

15.P

(bot

h r

ed o

r bo

th b

lack

)�2 55 1�

16.P

(bot

h a

ces

or b

oth

red

)� 25 25 1�

17.P

(bot

h 2

s or

bot

h l

ess

than

5)

� 21 21 1�18

.P(b

oth

bla

ck o

r bo

th l

ess

than

5)

�1 68 68 3�

For

Exe

rcis

es 1

9 an

d 2

0,u

se t

he

Ven

n d

iagr

am t

hat

sh

ows

the

nu

mb

er o

f p

arti

cip

ants

in

tw

o d

iffe

ren

t k

ind

s of

aer

obic

exe

rcis

e cl

asse

s th

at a

re o

ffer

ed a

t a

hea

lth

clu

b.

Det

erm

ine

each

pro

bab

ilit

y if

a p

erso

n

is s

elec

ted

at

ran

dom

fro

m t

he

par

tici

pan

ts.

19.P

(ste

p ae

robi

cs o

r ja

zzer

cise

,bu

t n

ot b

oth

)�4 69 2�

20.P

(ste

p ae

robi

cs a

nd

jazz

erci

se)

�1 63 2�

Jazz

erci

seS

tep

Aer

ob

ics

2722

13

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An

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con

tain

s 7

wh

ite

mar

ble

s an

d 5

blu

e m

arb

les.

Fou

r m

arb

les

are

sele

cted

wit

hou

t re

pla

cem

ent.

Fin

d e

ach

pro

bab

ilit

y.

1.P

(4 w

hit

e or

4 b

lue)

� 98 9�2.

P(e

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.Of

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take

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12-5

12-5



Readin

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12-5

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Lesson 12-5

Pre-

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-5 a

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e ev

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e ev

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incl

usi

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he

sho

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),w

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Pro

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Wh

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gam

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re c

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ce?

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fin

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e n

ine

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1 t

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in t

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e pr

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th

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o x

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cas

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ces

of w

inn

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1.F

ind

the

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gem

ents

in

wh

ich

B w

ins

and

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ann

ot.

o o

xo

x o

x o

x8

x o

x4

x x

ox

x o

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elow

are

12

of t

he

arra

nge

men

ts i

n w

hic

h A

win

s an

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can

not

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teth

e n

um

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to

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th

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flec

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tati

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for

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hat

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3.T

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diff

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d th

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r a

draw

or

for

A o

r B

to

win

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

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AT

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____

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12-5

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An

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tion

Sta

tist

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Mea

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NA

ME

____

____

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12-6

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Lesson 12-6

Mea

sure

s o

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entr

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end

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Use

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en

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sure

s o

fm

ean

the

data

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spr

ead

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t an

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of

valu

es

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tral

Ten

den

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edia

nth

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ta c

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in o

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Fin

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6.T

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tabl

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e po

pula

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e

six

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hic

h w

ould

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the

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t ap

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ntr

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lain

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ce: w

ww.fa

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pu

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tlie

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d

wo

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rai

se t

he

mea

n t

oo

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Lesson 12-6

Fin

d t

he

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d s

tan

dar

d d

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tion

of

each

set

of

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a to

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For

Exe

rcis

es 8

an

d 9

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th

e ta

ble

th

at s

how

s th

e p

rofi

t in

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lion

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orte

d b

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he

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om 1

997

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ce: U

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ensu

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edia

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dat

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t re

pres

ents

th

e da

ta?

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lain

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med

ian

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ore

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rese

nta

tive

bec

ause

th

e va

lue

45.3

is n

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clo

se t

oth

e o

ther

dat

a p

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nd

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wer

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ean

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he

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le t

hat

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ows

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cen

t of

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r ab

ove

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pro

fici

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n a

nat

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ally

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ssm

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ce: N

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nal C

ente

r for

Edu

catio

n St

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ics

10.F

ind

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ple

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t at

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2001

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ay.A

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min

istr

ator

,wou

ld y

ou q

uot

e th

e m

ean

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edia

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r m

ode

as a

n i

ndi

cato

r of

th

e am

oun

t of

dai

ly m

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tten

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ts o

nth

is f

loor

rec

eive

? E

xpla

in.

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her

th

e m

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r th

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od

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ey a

re e

qu

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nd

hig

her

th

an t

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h is

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ered

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the

smal

ler

amo

un

t o

f ti

me

the

phy

sici

ans

spen

d w

ith

th

e p

atie

nts

.

For

Exe

rcis

es 8

–10,

use

th

e fr

equ

ency

tab

le t

hat

sh

ows

the

per

cen

t of

pu

bli

c sc

hoo

lte

ach

ers

in t

he

U.S

.in

199

9 w

ho

use

d c

omp

ute

rs o

r th

eIn

tern

et a

t sc

hoo

l fo

r va

riou

sad

min

istr

ativ

e an

d t

each

ing

acti

viti

es.

8.F

ind

the

mea

n,m

edia

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ode

of t

he

data

.17

.75%

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9.S

uppo

se y

ou b

elie

ve t

each

ers

use

com

pute

rs o

r th

e In

tern

et t

ooin

freq

uen

tly.

Wh

ich

mea

sure

w

ould

you

quo

te a

s th

e “a

vera

ge?”

Sour

ce: N

atio

nal A

sses

smen

t of E

duca

tiona

l Pro

gres

s

Exp

lain

.M

od

e;it

is lo

wes

t.

10.S

upp

ose

you

bel

ieve

tea

cher

s u

se c

ompu

ters

or

the

Inte

rnet

too

oft

en.W

hic

h m

easu

rew

ould

you

qu

ote

as t

he

“ave

rage

?”E

xpla

in.

Mea

n;

it is

hig

hes

t.

For

Exe

rcis

es 1

1 an

d 1

2,u

se t

he

freq

uen

cy t

able

th

at

show

s th

e n

um

ber

of

gam

es p

laye

d b

y 24

Am

eric

an

Lea

gue

bas

ebal

l p

laye

rs b

etw

een

op

enin

g d

ay,2

001

and

Sep

tem

ber

8,2

001.

11.F

ind

the

mea

n,m

edia

n,m

ode,

and

stan

dard

dev

iati

on o

f th

en

um

ber

of g

ames

pla

yed

to t

he

nea

rest

ten

th.

138.

2,13

8;13

8,2.

0

12.F

or h

ow m

any

play

ers

is t

he

nu

mbe

r of

gam

es w

ith

in o

ne

stan

dard

dev

iati

on o

f th

e m

ean

?14

Sour

ce: M

ajor L

eagu

e Ba

seba

ll

No

.of

Gam

esF

req

uen

cy

141

4

140

3

139

4

138

5

137

2

136

3

135

3

Per

cen

t U

sin

g

Act

ivit

yC

om

pu

ter

or

Inte

rnet

Cre

ate

inst

ruct

iona

l mat

eria

ls39

Adm

inis

trat

ive

reco

rd k

eepi

ng34

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mun

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e w

ith c

olle

ague

s23

Gat

her

info

rmat

ion

for

plan

ning

less

ons

16

Mul

timed

ia c

lass

room

pre

sent

atio

ns8

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ess

rese

arch

and

bes

t pr

actic

es f

or t

each

ing

8

Com

mun

icat

e w

ith p

aren

ts o

r st

uden

ts8

Acc

ess

mod

el le

sson

pla

ns6

Pra

ctic

e (

Ave

rag

e)

Sta

tist

ical

Mea

sure

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

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____

__P

ER

IOD

____

_

12-6

12-6


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csS

tati

stic

al M

easu

res

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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ER

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_

12-6

12-6

©G

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3G

lenc

oe A

lgeb

ra 2

Lesson 12-6

Pre-

Act

ivit

yW

hat

sta

tist

ics

shou

ld a

tea

cher

tel

l th

e cl

ass

afte

r a

test

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-6 a

t th

e to

p of

pag

e 66

4 in

you

r te

xtbo

ok.

Th

ere

is m

ore

than

on

e w

ay t

o gi

ve a

n “

aver

age”

scor

e fo

r th

is t

est.

Th

ree

mea

sure

s of

cen

tral

ten

denc

y fo

r th

ese

scor

es a

re 9

4,76

.5 a

nd 7

3.9.

Can

you

tell

whi

ch o

f th

ese

is t

he m

ean,

the

med

ian,

and

the

mod

e w

itho

ut d

oing

any

calc

ula

tion

s? E

xpla

in y

our

answ

er.

Sam

ple

an

swer

:Yes

.Th

e m

od

e m

ust

be

on

e o

f th

e sc

ore

s,so

itm

ust

be

an in

teg

er.T

he

med

ian

mu

st b

e ei

ther

on

e o

f th

esc

ore

s o

r h

alfw

ay b

etw

een

tw

o o

f th

e sc

ore

s,so

it m

ust

be

anin

teg

er o

r a

dec

imal

en

din

g w

ith

.5.T

her

efo

re,9

4 is

th

e m

od

e,76

.5 is

th

e m

edia

n,a

nd

73.

9 is

th

e m

ean

.

Rea

din

g t

he

Less

on

1.M

atch

eac

h m

easu

re w

ith

on

e of

th

e si

x de

scri

ptio

ns

of h

ow t

o fi

nd

mea

sure

s of

cen

tral

ten

den

cy a

nd

vari

atio

n.

a.m

edia

nvi

b.m

ode

ic.

ran

geiv

d.

vari

ance

iiie.

mea

nii

f.st

anda

rd d

evia

tion

v

i.F

ind

the

mos

t co

mm

only

occ

urr

ing

valu

es o

r va

lues

in

a s

et o

f da

ta.

ii.A

dd t

he

data

an

d di

vide

by

the

nu

mbe

r of

ite

ms.

iii.

Fin

d th

e m

ean

of

the

squ

ares

of

the

diff

eren

ces

betw

een

eac

h v

alu

e in

th

e se

t of

dat

aan

d th

e m

ean

.

iv.

Fin

d th

e di

ffer

ence

bet

wee

n t

he

larg

est

and

smal

lest

val

ues

in

th

e se

t of

dat

a.

v.T

ake

the

posi

tive

squ

are

root

of

the

vari

ance

.

vi.I

f th

ere

is a

n o

dd n

um

ber

of i

tem

s in

a s

et o

f da

ta,t

ake

the

mid

dle

one.

If t

her

e is

an

even

nu

mbe

r of

ite

ms,

add

the

two

mid

dle

item

s an

d di

vide

by

2.

Hel

pin

g Y

ou

Rem

emb

er

2.It

is

usu

ally

eas

ier

to r

emem

ber

a co

mpl

icat

ed p

roce

dure

if

you

bre

ak i

t do

wn

in

to s

teps

.W

rite

th

e pr

oced

ure

for

fin

din

g th

e st

anda

rd d

evia

tion

for

a s

et o

f da

ta i

n a

ser

ies

ofbr

ief,

nu

mbe

red

step

s.

Sam

ple

an

swer

:1.

Fin

d t

he

mea

n.

2.F

ind

th

e d

iffe

ren

ce b

etw

een

eac

h v

alu

e an

d t

he

mea

n.

3.S

qu

are

each

dif

fere

nce

.4.

Fin

d t

he

mea

n o

f th

e sq

uar

es.

5.Ta

ke t

he

po

siti

ve s

qu

are

roo

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12-6

12-6

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ple

Exam

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Stu

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nte

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____

____

____

____

____

____

____

____

____

____

____

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____

____

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____

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12-7

12-7

Exam

ple

Exam

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Exer

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Exer

cises


An

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s


Skil

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____

____

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12-7

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Lesson 12-7

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wh

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able

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pos

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nor

ma

lly

dis

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1.2.

no

rmal

ly d

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neg

ativ

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skew

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an

d 4

,use

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tab

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sh

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aver

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the

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o th

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,use

th

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llow

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info

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he

tim

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cou

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wit

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mea

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min

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s an

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stan

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bout

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bout

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bout

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the

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,use

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the

nu

mb

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.

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data

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TEST

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For

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–10,

use

th

e fo

llow

ing

info

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he

scor

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min

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am

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of

100

and

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per

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0 an

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6.A

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hat

per

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peop

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the

test

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man

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th

an 1

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12-7

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Pre-

Act

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ow a

re t

he

hei

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thle

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trib

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trod

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on t

o L

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n 12

-7 a

t th

e to

p of

pag

e 67

1 in

you

r te

xtbo

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Th

ere

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pla

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on

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am a

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the

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bou

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hat

fra

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ayer

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72

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75,

incl

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an

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t �2 3�

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g t

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Less

on

1.In

dica

te w

het

her

eac

h o

f th

e fo

llow

ing

stat

emen

ts i

s tr

ue

or f

alse

.

a.In

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onti

nu

ous

prob

abil

ity

dist

ribu

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,th

ere

is a

fin

ite

nu

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pos

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tcom

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fals

e

b.

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al d

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be

repr

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bel

l cu

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tru

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c.A

dis

trib

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hat

is

repr

esen

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cu

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that

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hig

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e ri

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egat

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lse

d.

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ske

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fals

e

2.M

s.R

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gave

the

sam

e qu

iz t

o he

r tw

o ge

omet

ry c

lass

es.S

he r

ecor

ded

the

follo

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g sc

ores

.

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st-p

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d c

lass

:

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th-p

erio

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:

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ach

cla

ss,3

0 st

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nts

too

k th

e qu

iz.T

he

mea

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lass

was

6.4

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ich

set

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core

s h

as t

he

grea

ter

stan

dard

dev

iati

on?

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swer

th

is q

ues

tion

wit

hou

t do

ing

any

calc

ula

tion

s.)

Exp

lain

you

r an

swer

.

Fir

st p

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;sa

mp

le a

nsw

er:T

he

sco

res

are

mo

re s

pre

ad o

ut

fro

mth

e m

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th

an f

or

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fift

h p

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Rem

emb

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3.M

any

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den

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if a

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mem

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this

?

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ollo

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f th

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on

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ft(n

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),th

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neg

ativ

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ng

nor

th f

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th

e in

ters

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f F

irst

Ave

nu

e an

d F

irst

S

tree

t,th

e av

enu

es a

re 1

st,2

nd,

3rd,

and

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n.G

oin

g ea

st,t

he

stre

ets

are

nu

mbe

red

in t

he

sam

e w

ay.

Fact

oria

ls c

an b

e u

sed

to f

ind

the

nu

mbe

r,r(

e,n

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di

ffer

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o in

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how

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elow

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r(e,

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�

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e;t

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umbe

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enue

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nort

h is

n.

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e fo

llow

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prob

lem

s ex

amin

e th

e po

ssib

le r

oute

s fr

om o

ne

loca

tion

to

an

oth

er.A

ssu

me

that

you

nev

er u

se a

rou

te t

hat

is

un

nec

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rily

lon

g.A

ssu

me

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e

1 an

d n

1.

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ve e

ach

pro

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m.

1.L

ist

all

the

poss

ible

rou

tes

from

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Str

eet

and

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nu

e to

4th

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eet

and

3rd

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nu

e.U

se o

rder

ed p

airs

to

show

th

e ro

ute

s,w

ith

str

eet

nu

mbe

rs f

irst

,an

d av

enu

e n

um

bers

sec

ond.

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mpl

e,ea

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oute

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t (1

,1)

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at

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(4,1

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(3,1

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(2,2

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(3,2

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(3,2

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(2,3

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�(1

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,3)

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,3)

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�(4

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2.U

se t

he

form

ula

to

com

pute

th

e n

um

ber

of r

oute

s fr

om (

1,1)

to

(4,3

).T

her

e ar

e 4

stre

ets

goin

g ea

st a

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3 av

enu

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oin

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orth

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2nd St.

3rd St.

4th St.

5th St.

6th St.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

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12-7

12-7


An

swer

s


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12-8

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Bin

om

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xpan

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For

sit

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ou c

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Bin

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o fi

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prob

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s.T

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term

s in

a b

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expa

nsi

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e fo

un

d by

usi

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bin

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rob

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that

3 c

oin

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d 3

sh

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ails

wh

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coi

ns

are

toss

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ere

are

2 po

ssib

le o

utc

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th

at a

re e

qual

ly l

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y:h

eads

(H

) an

d ta

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e to

sses

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6 co

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are

inde

pen

den

t ev

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.Wh

en (

H �

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is e

xpan

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the

term

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tain

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H3 T

3 ,w

hic

h r

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nd

3 ta

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is u

sed

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et t

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desi

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prob

abil

ity.

By

the

Bin

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th

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or 0

.312

5.

Fin

d e

ach

pro

bab

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)

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s on

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tru

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d f

alse

are

even

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,fin

d e

ach

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t 3

corr

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answ

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or

0.04

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r 0.

172

7.A

die

is

toss

ed 4

tim

es.W

hat

is

the

prob

abil

ity

of t

ossi

ng

exac

tly

two

sixe

s?

or

0.11

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mia

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fix

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he t

rials

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he p

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bilit

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the

sam

e.

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a co

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s w

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ted

so

that

th

e p

rob

abil

ity

of g

etti

ng

hea

ds

inan

y on

e to

ss i

s 90

%.W

hat

is

the

pro

bab

ilit

y of

get

tin

g ex

actl

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tos

ses?

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e pr

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nd

the

prob

abil

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tail

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her

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bou

t 38

%.

1.B

ASK

ETB

ALL

For

an

y on

e fo

ul

shot

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ek h

as a

pro

babi

lity

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0.72

of

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ing

the

shot

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he

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s pa

rt o

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prac

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dri

ll,h

e sh

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hot

s fr

om t

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fou

l li

ne.

a.W

hat

is

the

prob

abil

ity

that

he

gets

in

exa

ctly

6 f

oul

shot

s? a

bo

ut

31%

b.W

hat

is

the

prob

abil

ity

that

he

gets

in

at

leas

t 6

fou

l sh

ots?

ab

ou

t 60

%

2.SC

HO

OL

A t

each

er i

s tr

yin

g to

dec

ide

wh

eth

er t

o h

ave

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ques

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mu

ltip

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hoi

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est.

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e w

ants

to

prev

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stu

den

ts w

ho

just

gu

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sco

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ell

on t

he

test

.

a.O

n a

5-q

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mu

ltip

le-c

hoi

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est

wit

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ch

oice

s pe

r qu

esti

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hat

is

the

prob

abil

ity

that

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tude

nt

can

sco

re a

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ast

60%

by

gues

sin

g? 1

0.4%

b.W

hat

is

the

prob

abil

ity

that

a s

tude

nt

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sco

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by

gues

sin

g on

a t

est

ofth

e sa

me

len

gth

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h 5

ch

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s pe

r qu

esti

on?

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lie

roll

s tw

o di

ce a

nd

adds

th

e tw

o n

um

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.

a.W

hat

is

the

prob

abil

ity

that

th

e su

m w

ill

be d

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by

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f sh

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sum

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e di

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? ab

ou

t 16

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ATI

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ter

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% o

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wh

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trip

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rin

g on

e pr

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essi

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e at

tem

pts

20 t

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els.

a.W

hat

is

the

prob

abil

ity

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wil

l fa

ll o

nly

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ce?

abo

ut

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b.W

hat

is

the

prob

abil

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l fa

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es?

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ut

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____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-8

12-8

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Bin

om

ial E

xper

imen

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-8

12-8

©G

lenc

oe/M

cGra

w-H

ill74

3G

lenc

oe A

lgeb

ra 2

Lesson 12-8

Fin

d e

ach

pro

bab

ilit

y if

a c

oin

is

toss

ed 4

tim

es.

1.P

(4 h

eads

)� 11 6�

2.P

(0 h

eads

)� 11 6�

3.P

(exa

ctly

3 h

eads

)�1 4�

4.P

(exa

ctly

2 h

eads

)�3 8�

5.P

(exa

ctly

1 h

ead)

�1 4�6.

P(a

t le

ast

3 h

eads

)� 15 6�

Fin

d e

ach

pro

bab

ilit

y if

a d

ie i

s ro

lled

3 t

imes

.

7.P

(exa

ctly

on

e 2)

�2 75 2�8.

P(e

xact

ly t

wo

2s)

� 75 2�

9.P

(exa

ctly

th

ree

2s)

� 21 16�10

.P(a

t m

ost

one

2)�2 25 7�

A t

own

th

at p

rese

nts

a f

irew

ork

s d

isp

lay

du

rin

g it

s J

uly

4 c

eleb

rati

on f

oun

d t

he

pro

bab

ilit

y th

at a

fam

ily

wit

h t

wo

or m

ore

chil

dre

n w

ill

wat

ch t

he

fire

wor

ks

is �3 5� .

If 5

of

thes

e fa

mil

ies

are

sele

cted

at

ran

dom

,fin

d e

ach

pro

bab

ilit

y.

11.P

(exa

ctly

3 f

amil

ies

wat

ch t

he

fire

wor

ks)

12.P

(exa

ctly

2 f

amil

ies

wat

ch t

he

fire

wor

ks)

�2 61 26 5��1 64 24 5�

13.P

(exa

ctly

5 f

amil

ies

wat

ch t

he

fire

wor

ks)

14.P

(no

fam

ilie

s w

atch

th

e fi

rew

orks

)

� 32 14 23 5�

� 33 12 25�

15.P

(at

leas

t 4

fam

ilie

s w

atch

th

e fi

rew

orks

)16

.P(a

t m

ost

1 fa

mil

y w

atch

es t

he

fire

wor

ks)

�1 30 15 23 5�

� 32 17 22 5�

On

e se

ctio

n o

f a

stan

dar

diz

ed E

ngl

ish

lan

guag

e te

st h

as 1

0 tr

ue/

fals

e q

ues

tion

s.F

ind

eac

h p

rob

abil

ity

wh

en a

stu

den

t gu

esse

s at

all

ten

qu

esti

ons.

17.P

(exa

ctly

8 c

orre

ct)

� 14 05 24�18

.P(e

xact

ly 2

cor

rect

)� 14 05 24�

19.P

(exa

ctly

hal

f co

rrec

t)� 26 53 6�

20.P

(all

10

corr

ect)

� 101 24�

21.P

(0 c

orre

ct)

� 101 24�

22.P

(at

leas

t 8

corr

ect)

� 17 28�

©G

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cGra

w-H

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4G

lenc

oe A

lgeb

ra 2

Fin

d e

ach

pro

bab

ilit

y if

a c

oin

is

toss

ed 6

tim

es.

1.P

(exa

ctly

3 t

ails

)� 15 6�

2.P

(exa

ctly

5 t

ails

)� 33 2�

3.P

(0 t

ails

)� 61 4�

4.P

(at

leas

t 4

hea

ds)

�1 31 2�

5.P

(at

leas

t 4

tail

s)�1 31 2�

6.P

(at

mos

t 2

tail

s)�1 31 2�

Th

e p

rob

abil

ity

of C

hri

s m

akin

g a

free

th

row

is

�2 3� .If

sh

e sh

oots

5 t

imes

,fin

d e

ach

pro

bab

ilit

y.

7.P

(all

mis

sed)

� 21 43�8.

P(a

ll m

ade)

� 23 42 3�

9.P

(exa

ctly

2 m

ade)

� 24 40 3�10

.P(e

xact

ly 1

mis

sed)

� 28 40 3�

11.P

(at

leas

t 3

mad

e)�6 84 1�

12.P

(at

mos

t 2

mad

e)�1 87 1�

Wh

en T

arin

an

d S

am p

lay

a ce

rtai

n b

oard

gam

e,th

e p

rob

abil

ity

that

Tar

in w

ill

win

a

gam

e is

�3 4� .If

th

ey p

lay

5 ga

mes

,fin

d e

ach

pro

bab

ilit

y.

13.P

(Sam

win

s on

ly o

nce

)� 14 00 25 4

�14

.P(T

arin

win

s ex

actl

y tw

ice)

� 54 15 2�

15.P

(Sam

win

s ex

actl

y 3

gam

es)

� 54 15 2�16

.P(S

am w

ins

at l

east

1 g

ame)

� 17 08 21 4�

17.P

(Tar

in w

ins

at l

east

3 g

ames

)�4 55 19 2�

18.P

(Tar

in w

ins

at m

ost

2 ga

mes

)� 55 13 2�

19.S

AFE

TYIn

Au

gust

200

1,th

e A

mer

ican

Au

tom

obil

e A

ssoc

iati

on r

epor

ted

that

73%

of

Am

eric

ans

use

sea

t be

lts.

In a

ran

dom

sel

ecti

on o

f 10

Am

eric

ans

in 2

001,

wh

at i

s th

epr

obab

ilit

y th

at e

xact

ly h

alf

of t

hem

use

sea

t be

lts?

Sour

ce:A

AAab

ou

t 7.

5%

HEA

LTH

For

Exe

rcis

es 2

0 an

d 2

1,u

se t

he

foll

owin

g in

form

atio

n.

In 2

001,

the

Am

eric

an H

eart

Ass

ocia

tion

rep

orte

d th

at 5

0 pe

rcen

t of

th

e A

mer

ican

s w

ho

rece

ive

hear

t tr

ansp

lant

s ar

e ag

es 5

0–64

and

20

perc

ent

are

ages

35–

49.

Sour

ce: A

mer

ican

Hear

t Ass

ociat

ion

20.I

n a

ran

dom

ly s

elec

ted

grou

p of

10

hea

rt t

ran

spla

nt

reci

pien

ts,w

hat

is

the

prob

abil

ity

that

at

leas

t 8

of t

hem

are

age

s 50

–64?

� 17 28�

21.I

n a

ran

dom

ly s

elec

ted

grou

p of

5 h

eart

tra

nsp

lan

t re

cipi

ents

,wh

at i

s th

e pr

obab

ilit

yth

at 2

of

them

are

age

s 35

–49?

�1 62 28 5�

Pra

ctic

e (

Ave

rag

e)

Bin

om

ial E

xper

imen

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-8

12-8


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csB

ino

mia

l Exp

erim

ents

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-8

12-8

©G

lenc

oe/M

cGra

w-H

ill74

5G

lenc

oe A

lgeb

ra 2

Lesson 12-8

Pre-

Act

ivit

yH

ow c

an y

ou d

eter

min

e w

het

her

gu

essi

ng

is w

orth

it?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-8 a

t th

e to

p of

pag

e 67

6 in

you

r te

xtbo

ok.

Su

ppos

e yo

u a

re t

akin

g a

50-q

ues

tion

mu

ltip

le-c

hoi

ce t

est

in w

hic

h t

her

ear

e 5

answ

er c

hoi

ces

for

each

qu

esti

on.Y

ou a

re t

old

that

no

poin

ts w

ill

bede

duct

ed f

or w

ron

g an

swer

s.S

hou

ld y

ou g

ues

s th

e an

swer

s to

th

e qu

esti

ons

you

do

not

kn

ow?

Exp

lain

you

r re

ason

ing.

Sam

ple

an

swer

:Yes

;th

e p

rob

abili

ty o

f g

ues

sin

g t

he

rig

ht

answ

er t

o a

qu

esti

on

is �1 5� ,

so

you

hav

e a

chan

ce t

o g

et s

om

e p

oin

ts b

y g

ues

sin

g,a

nd

yo

uh

ave

no

thin

g t

o lo

se.

Rea

din

g t

he

Less

on

1.In

dica

te w

het

her

eac

h o

f th

e fo

llow

ing

is a

bin

omia

l ex

peri

men

tor

not

a b

inom

ial

expe

rim

ent.

If t

he

expe

rim

ent

is n

ot a

bin

omia

l ex

peri

men

t,ex

plai

n w

hy.

a.A

fai

r co

in i

s to

ssed

10

tim

es a

nd

“hea

ds”

or “

tail

s”is

rec

orde

d ea

ch t

ime.

bin

om

ial

exp

erim

ent

b.

A p

air

of d

ice

is t

hro

wn

5 t

imes

an

d th

e su

m o

f th

e n

um

bers

th

at c

ome

up

is r

ecor

ded

each

tim

e.N

ot

a b

ino

mia

l exp

erim

ent;

ther

e ar

e m

ore

th

an t

wo

po

ssib

leo

utc

om

es f

or

each

tri

al.

c.T

her

e ar

e 5

red

mar

bles

an

d 6

blu

e m

arbl

es i

n a

bag

.On

e m

arbl

e is

dra

wn

fro

m t

he

bag

and

its

colo

r re

cord

ed.T

he

mar

ble

is n

ot p

ut

back

in

th

e ba

g.A

sec

ond

mar

ble

isdr

awn

an

d it

s co

lor

reco

rded

.N

ot

a b

ino

mia

l exp

erim

ent;

the

tria

ls a

re n

ot

ind

epen

den

t (o

r,th

e p

rob

abili

ties

fo

r th

e tw

o t

rial

s ar

e n

ot

the

sam

e).

d.

Th

ere

are

5 re

d m

arbl

es a

nd

6 bl

ue

mar

bles

in

a b

ag.O

ne

mar

ble

is d

raw

n f

rom

th

eba

g an

d it

s co

lor

reco

rded

.Th

e m

arbl

e is

pu

t ba

ck i

n t

he

bag.

A s

econ

d m

arbl

e is

draw

n a

nd

its

colo

r re

cord

ed.

bin

om

ial e

xper

imen

t

2.L

en r

ando

mly

gu

esse

s th

e an

swer

s to

all

6 m

ult

iple

-ch

oice

qu

esti

ons

on h

is c

hem

istr

yte

st.E

ach

qu

esti

on h

as 5

ch

oice

s.W

hic

h o

f th

e fo

llow

ing

expr

essi

ons

give

s th

epr

obab

ilit

y th

at h

e w

ill

get

at l

east

4 o

f th

e an

swer

s co

rrec

t?B

A.

P(6

,4) ��1 5� �4 ��4 5� �2

�P

(6,5

) ��1 5� �5 ��4 5� �1�

P(6

,6) ��1 5� �6 ��4 5� �0

B.C

(6,4

) ��1 5� �4 ��4 5� �2�

C(6

,5) ��1 5� �5 ��4 5� �1

�C

(6,6

) ��1 5� �6 ��4 5� �0

C.

C(6

,4) ��1 5� �2 ��4 5� �4

�C

(6,5

) ��1 5� �1 ��4 5� �5�

C(6

,6) ��1 5� �0 ��4 5� �6

Hel

pin

g Y

ou

Rem

emb

er3.

Som

e st

uden

ts h

ave

trou

ble

rem

embe

ring

how

to

calc

ulat

e bi

nom

ial

prob

abil

itie

s.W

hat

is

an e

asy

way

to

rem

embe

r w

hic

h n

um

bers

to

put

into

an

exp

ress

ion

lik

e C

(6,4

) ��1 5� �2 ��4 5� �4?

Sam

ple

an

swer

:Th

e b

ino

mia

l co

effi

cien

t is

C(n

,r),

wh

ere

nis

th

e n

um

ber

of

tria

ls a

nd

ris

th

e n

um

ber

of

succ

esse

s.T

he

pro

bab

ility

of

succ

ess

isra

ised

to

th

e rt

h p

ow

er a

nd

th

e p

rob

abili

ty o

f fa

ilure

is r

aise

d t

o t

he

(n�

r)th

po

wer

.

©G

lenc

oe/M

cGra

w-H

ill74

6G

lenc

oe A

lgeb

ra 2

Mis

use

s o

f S

tati

stic

sS

tati

stic

s ca

n b

e m

isle

adin

g.G

raph

s fo

r a

set

of d

ata

can

loo

k ve

ry d

iffe

ren

tfr

om o

ne

anot

her

.Com

pare

th

e fo

llow

ing

grap

hs.

Not

ice

that

th

e tw

o gr

aph

s sh

ow t

he

sam

e da

ta,b

ut

the

spac

ing

in t

he

vert

ical

an

d h

oriz

onta

l sc

ales

dif

fers

.Sca

les

can

be

cram

ped

or s

prea

d ou

t to

mak

e a

grap

h t

hat

giv

es a

cer

tain

im

pres

sion

.Wh

ich

gra

ph w

ould

you

use

to

give

th

e im

pres

sion

th

at t

he

un

empl

oym

ent

rate

dro

pped

dra

mat

ical

ly f

rom

1990

to

2000

?th

e se

con

d g

rap

h

Su

ppos

e th

at a

car

com

pan

y cl

aim

s,“7

5% o

f pe

ople

su

rvey

ed s

ay t

hat

ou

r ca

ris

bet

ter

than

th

e co

mpe

titi

on.”

If f

our

peop

le w

ere

aske

d w

hic

h c

ar t

hey

pref

erre

d an

d 75

% a

gree

d,h

ow m

any

peop

le t

hou

ght

that

Ou

r C

arw

asbe

tter

?3

peo

ple

Th

e ad

vert

isem

ent

was

mis

lead

ing

in o

ther

way

s as

wel

l.F

or e

xam

ple,

wh

ow

as s

urv

eyed

—w

ere

the

peop

le c

ompa

ny

empl

oyee

s,or

im

part

ial

buye

rs?

Su

pp

ose

an a

dve

rtis

er c

laim

s th

at 9

0% o

f al

l of

on

e b

ran

d o

f ca

r so

ldin

th

e la

st 1

0 ye

ars

are

stil

l on

th

e ro

ad.

1.If

10,

000

cars

wer

e so

ld,h

ow m

any

are

stil

l on

th

e ro

ad?

9,00

0

2.If

100

0 ca

rs w

ere

sold

,how

man

y ar

e st

ill

on t

he

road

?90

0

3.F

ind

an e

xam

ple

to s

how

how

you

th

ink

aver

ages

cou

ld b

e u

sed

in a

mis

lead

ing

way

.S

ee s

tud

ents

’wo

rk.

4.A

su

rvey

of

a la

rge

sam

ple

of p

eopl

e w

ho

own

sm

all

com

pute

rs r

evea

led

that

85%

of

the

peop

le t

hou

ght

the

inst

ruct

ion

man

ual

s sh

ould

be

bett

erw

ritt

en.A

man

ufa

ctu

rer

of s

mal

l co

mpu

ters

cla

imed

th

at i

t su

rvey

edm

any

of t

he

sam

e pe

ople

an

d fo

un

d th

at a

ll o

f th

em l

iked

th

eir

man

ual

s.D

iscu

ss t

he

poss

ible

dis

crep

ancy

in

th

e re

sult

s.S

ee s

tud

ents

’wo

rk.

U.S

. Un

emp

loym

ent

Rat

e

Year

Percent

0’9

0’9

2’9

4’9

6’0

2’9

8’0

0

8 7 6 5 4

Sour

ce: U

.S. D

epar

tmen

t of L

abor

U.S

. Un

emp

loym

ent

Rat

e

Year

Percent

0’9

0’9

2’9

4’9

6’0

2’9

8’0

0’9

1’9

3’9

5’9

7’9

9’0

1

8 7 6 5 4

Sour

ce: U

.S. D

epar

tmen

t of L

abor

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-8

12-8



Stu

dy G

uid

e a

nd I

nte

rven

tion

Sam

plin

g a

nd

Err

or

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-9

12-9

©G

lenc

oe/M

cGra

w-H

ill74

7G

lenc

oe A

lgeb

ra 2

Lesson 12-9

Bia

sA

sam

ple

of s

ize

nis

ran

dom

(or

un

bia

sed

) w

hen

eve

ry p

ossi

ble

sam

ple

of s

ize

nh

asan

equ

al c

han

ce o

f be

ing

sele

cted

.If

a sa

mpl

e is

bia

sed,

then

in

form

atio

n o

btai

ned

fro

m i

tm

ay n

ot b

e re

liab

le.

To

fin

d o

ut

how

peo

ple

in

th

e U

.S.f

eel

abou

t m

ass

tran

sit,

peo

ple

at

a co

mm

ute

r tr

ain

sta

tion

are

ask

ed t

hei

r op

inio

n.D

oes

this

sit

uat

ion

rep

rese

nt

ara

nd

om s

amp

le?

No;

the

sam

ple

incl

ude

s on

ly p

eopl

e w

ho

actu

ally

use

a m

ass-

tran

sit

faci

lity

.Th

e sa

mpl

edo

es n

ot i

ncl

ude

peo

ple

wh

o ri

de b

ikes

,dri

ve c

ars,

or w

alk.

Det

erm

ine

wh

eth

er e

ach

sit

uat

ion

wou

ld p

rod

uce

a r

and

om s

amp

le.W

rite

yes

orn

oan

d e

xpla

in y

our

answ

er.

1.as

kin

g pe

ople

in

Ph

oen

ix,A

rizo

na,

abou

t ra

infa

ll t

o de

term

ine

the

aver

age

rain

fall

for

the

Un

ited

Sta

tes

No

;it

rai

ns

less

in P

ho

enix

th

an m

ost

pla

ces

in t

he

U.S

.

2.ob

tain

ing

the

nam

es o

f tr

ee t

ypes

in

Nor

th A

mer

ica

by s

urv

eyin

g al

l of

th

e U

.S.N

atio

nal

For

ests

Yes

;th

ere

are

Nat

ion

al F

ore

sts

in a

bo

ut

ever

y st

ate

in t

he

U.S

.

3.su

rvey

ing

ever

y te

nth

per

son

wh

o en

ters

th

e m

all

to f

ind

out

abou

t m

usi

c pr

efer

ence

s in

that

par

t of

th

e co

un

try

Yes;

mal

l cu

sto

mer

s sh

ou

ld b

e fa

irly

rep

rese

nta

tive

in t

erm

s o

f m

usi

c ta

stes

.

4.in

terv

iew

ing

cou

ntr

y cl

ub

mem

bers

to

dete

rmin

e th

e av

erag

e n

um

ber

of t

elev

isio

ns

per

hou

seh

old

in t

he

com

mu

nit

y N

o;

cou

ntr

y cl

ub

mem

ber

s w

ou

ld t

end

to

be

mo

re a

fflu

ent

and

th

us

no

t a

rep

rese

nta

tive

sam

ple

of

the

com

mu

nit

y.

5.su

rvey

ing

all

stu

den

ts w

hos

e ID

nu

mbe

rs e

nd

in 4

abo

ut

thei

r gr

ades

an

d ca

reer

coun

seli

ng n

eeds

Yes

;ID

nu

mb

ers

are

pro

bab

ly a

ssig

ned

alp

hab

etic

ally

or

byso

me

othe

r m

etho

d no

t co

nnec

ted

to s

tude

nts’

grad

es o

r co

unse

ling

need

s.

6.su

rvey

ing

pare

nts

at

a da

y ca

re f

acil

ity

abou

t th

eir

pref

eren

ces

for

bran

ds o

f ba

by f

ood

for

a m

arke

tin

g ca

mpa

ign

Yes

;ch

oic

e o

f a

day

care

fac

ility

wo

uld

pro

bab

lyn

ot

infl

uen

ce b

aby

foo

d p

refe

ren

ces.

7.as

kin

g pe

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ibra

ry a

bou

t th

e n

um

ber

of m

agaz

ines

to

wh

ich

th

ey s

ubs

crib

e in

orde

r to

des

crib

e th

e re

adin

g h

abit

s of

a t

own

No

;lib

rary

vis

ito

rs t

end

to

rea

dm

ore

th

an m

ost

cit

izen

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wh

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er e

ach

sit

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ion

wou

ld p

rod

uce

a r

and

om s

amp

le.W

rite

yes

orn

oan

d e

xpla

in y

our

answ

er.

1.ca

llin

g h

ouse

hol

ds a

t 3:

30 P

.M.o

n T

ues

day

to d

eter

min

e a

poli

tica

l ca

ndi

date

’s s

upp

ort

No

;si

nce

mo

st r

egis

tere

d v

ote

rs a

re li

kely

to

be

at w

ork

at

this

tim

e,th

issa

mp

le w

ou

ld n

ot

be

rep

rese

nta

tive

of

all r

egis

tere

d v

ote

rs.

2.po

llin

g cu

stom

ers

as t

hey

exi

t a

spor

tin

g go

ods

stor

e ab

out

thei

r at

titu

des

abou

t ex

erci

seN

o;

thes

e cu

sto

mer

s ar

e lik

ely

to v

alu

e ex

erci

se m

ore

th

an t

ho

se w

ho

do

no

t sh

op

at

spo

rtin

g g

oo

ds

sto

res,

wh

o a

re n

ot

rep

rese

nte

d in

th

is s

urv

ey.

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cord

ing

the

nu

mbe

r of

sit

-ups

per

form

ed b

y 15

-yea

r ol

d gi

rls

in t

he

hig

h s

choo

ls o

f a

larg

e sc

hoo

l di

stri

ct t

o de

term

ine

the

fitn

ess

of a

ll h

igh

-sch

ool

girl

s in

th

e di

stri

ctN

o;

15-y

ear

old

gir

ls m

ay n

ot

hav

e th

e sa

me

abili

ties

as

18-y

ear

old

sen

iors

,fo

r ex

amp

le.

4.se

lect

ing

two

of a

cit

y’s

20 a

part

men

t bu

ildi

ngs

for

a s

urv

ey t

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term

ine

the

desi

re o

fap

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ent

dwel

lers

in

th

e ci

ty t

o ow

n a

hom

eN

o;

the

resi

den

ts o

f th

e tw

obu

ildin

gs

sele

cted

mig

ht,

for

exam

ple

,hav

e n

icer

ap

artm

ents

or

be

in a

nic

er a

rea

of

tow

n,a

nd

th

us

wo

uld

no

t w

ell r

epre

sen

t th

e d

esir

es o

fp

eop

le in

oth

er b

uild

ing

s.

5.In

a l

arge

sch

ool

dist

rict

,th

e su

peri

nte

nde

nt

of s

choo

ls i

nte

rvie

ws

two

teac

her

s at

ran

dom

fro

m e

ach

sch

ool

to d

eter

min

e w

het

her

tea

cher

s in

th

e di

stri

ct t

hin

k st

ude

nts

are

assi

gned

too

mu

ch o

r to

o li

ttle

hom

ewor

k.Ye

s;si

nce

a c

ross

sec

tio

n o

fte

ach

ers

fro

m a

ll le

vels

was

sel

ecte

d a

t ra

nd

om

,th

e sa

mp

le s

ho

uld

wel

lre

pre

sen

t th

e p

op

ula

tio

n o

f te

ach

ers

in t

he

dis

tric

t.

6.F

or s

even

con

secu

tive

day

s,on

e h

our

each

in

th

e m

orn

ing,

afte

rnoo

n,a

nd

even

ing,

ever

yte

nth

cu

stom

er w

ho

ente

rs a

mal

l is

ask

ed t

o ch

oose

her

or

his

fav

orit

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ore.

Yes;

bec

ause

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e sa

mp

le is

ch

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le w

eek,

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rin

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ause

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e se

lect

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ou

ld w

ell r

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sen

t th

e g

ener

al p

op

ula

tio

nth

at s

ho

ps

at t

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mal

l sto

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67%

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thos

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rvey

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aid

they

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tri

edto

qu

it s

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wit

hin

th

e la

st y

ear.

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e m

argi

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ror

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3%

.Abo

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ype

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ld p

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and

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amp

le.W

rite

yes

orn

oan

d e

xpla

in y

our

answ

er.

1.ca

llin

g ev

ery

twen

tiet

h r

egis

tere

d vo

ter

to d

eter

min

e w

het

her

peo

ple

own

or

ren

t th

eir

hom

es i

n y

our

com

mu

nit

yN

o;

reg

iste

red

vo

ters

may

be

mo

re li

kely

to

be

ho

meo

wn

ers,

cau

sin

g t

he

surv

ey t

o u

nd

erre

pre

sen

t re

nte

rs.

2.pr

edic

tin

g lo

cal

elec

tion

res

ult

s by

pol

lin

g pe

ople

in

eve

ry t

wen

tiet

h r

esid

ence

in

all

th

edi

ffer

ent

nei

ghbo

rhoo

ds o

f yo

ur

com

mu

nit

yYe

s;si

nce

all

nei

gh

bo

rho

od

s ar

ere

pre

sen

ted

pro

po

rtio

nal

ly,t

he

view

s o

f th

e co

mm

un

ity

sho

uld

as

aw

ho

le s

ho

uld

be

wel

l rep

rese

nte

d.

3.to

fin

d ou

t w

hy

not

man

y st

ude

nts

are

usi

ng

the

libr

ary,

a sc

hoo

l’s l

ibra

rian

giv

es a

ques

tion

nai

re t

o ev

ery

ten

th s

tude

nt

ente

rin

g th

e li

brar

yN

o;

she

is p

olli

ng

on

lyth

e st

ud

ents

wh

o a

re c

om

ing

to

th

e lib

rary

,an

d w

ill o

bta

in n

o in

pu

t fr

om

tho

se w

ho

are

n’t

usi

ng

th

e lib

rary

.4.

test

ing

over

all

perf

orm

ance

of

tire

s on

in

ters

tate

hig

hw

ays

only

No

;fo

r ov

eral

lp

erfo

rman

ce,t

ires

sh

ou

ld b

e te

sted

on

man

y ki

nd

s o

f su

rfac

es,a

nd

un

der

man

y ty

pes

of

con

dit

ion

s.5.

sele

ctin

g ev

ery

50th

ham

burg

er f

rom

a f

ast-

food

res

tau

ran

t ch

ain

an

d de

term

inin

g it

sfa

t co

nte

nt

to a

sses

s th

e fa

t co

nte

nt

of h

ambu

rger

s se

rved

in

fas

t-fo

od r

esta

ura

nt

chai

ns

thro

ugh

out

the

cou

ntr

yN

o;

the

sele

cted

ham

burg

ers

are

a ra

nd

om

sam

ple

of

the

ham

burg

ers

serv

ed in

on

e ch

ain

,an

d m

ay r

epre

sen

t th

e fa

t co

nte

nt

for

that

ch

ain

,bu

t w

ill n

ot

nec

essa

rily

rep

rese

nt

the

fat

con

ten

t o

fh

ambu

rger

s se

rved

in o

ther

fas

t-fo

od

res

tau

ran

t ch

ain

s.6.

assi

gnin

g al

l sh

ift

wor

kers

in

a m

anu

fact

uri

ng

plan

t a

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iqu

e id

enti

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nu

mbe

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dth

en p

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nu

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hat

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g 30

at

ran

dom

to

dete

rmin

e th

e an

nu

alav

erag

e sa

lary

of

the

wor

kers

Yes;

bec

ause

th

e n

um

ber

s ar

e ra

nd

om

ly c

ho

sen

fro

m a

mo

ng

all

shif

t w

ork

ers,

all w

ork

ers

hav

e th

e sa

me

chan

ce o

f b

ein

gse

lect

ed.

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d t

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12-9

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Pre-

Act

ivit

yH

ow a

re o

pin

ion

pol

ls u

sed

in

pol

itic

al c

amp

aign

s?

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d th

e in

trod

ucti

on t

o L

esso

n 12

-9 a

t th

e to

p of

pag

e 68

2 in

you

r te

xtbo

ok.

Do

you

th

ink

the

resu

lts

of t

he

surv

ey a

bou

t th

e pr

esid

enti

al p

refe

ren

ces

dem

onst

rate

s th

at B

ush

was

act

ual

ly a

hea

d in

Flo

rida

a m

onth

bef

ore

the

elec

tion

? If

th

ere

is n

ot e

nou

gh i

nfo

rmat

ion

giv

en t

o de

term

ine

this

,lis

t at

leas

t tw

o qu

esti

ons

you

wou

ld a

sk a

bou

t th

e su

rvey

th

at w

ould

hel

p yo

ude

term

ine

the

sign

ific

ance

of

the

surv

ey.

Sam

ple

an

swer

:Th

ere

is n

ot

eno

ug

h in

form

atio

n t

o t

ell.

1.H

ow m

any

peo

ple

wer

e su

rvey

ed?

2.H

ow

was

th

e sa

mp

le f

or

the

surv

ey s

elec

ted

? 3

.Wh

at is

th

em

arg

in o

f er

ror

for

this

su

rvey

?

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h s

itu

atio

n w

ould

pro

duce

a r

ando

m s

ampl

e.W

rite

yes

or n

oan

dex

plai

n y

our

answ

er.

a.as

kin

g al

l th

e cu

stom

ers

at f

ive

rest

aura

nts

on

th

e sa

me

even

ing

how

man

y ti

mes

am

onth

th

ey e

at d

inn

er i

n r

esta

ura

nts

to

dete

rmin

e h

ow o

ften

th

e av

erag

e A

mer

ican

eats

din

ner

in

a r

esta

ura

nts

No

;p

eop

le s

urv

eyed

at

a re

stau

ran

t m

igh

t b

elik

ely

to e

at d

inn

er in

res

tau

ran

ts m

ore

oft

en t

han

oth

er p

eop

le.

b.

putt

ing

the

nam

es o

f al

l sen

iors

at

your

hig

h sc

hool

in a

hat

and

the

n dr

awin

g 20

nam

esfo

r a

surv

ey t

o fi

nd

out

wh

ere

sen

iors

wou

ld l

ike

to h

old

thei

r pr

omYe

s;ev

ery

sen

ior

wo

uld

hav

e an

eq

ual

ch

ance

of

bei

ng

ch

ose

n f

or

the

surv

ey.

2.A

sur

vey

dete

rmin

ed t

hat

58%

of

regi

ster

ed v

oter

s in

the

Uni

ted

Sta

tes

supp

ort

incr

ease

dfe

dera

l sp

endi

ng f

or e

duca

tion

.The

mar

gin

of e

rror

for

thi

s su

rvey

is

4%.E

xpla

in i

n yo

urow

n w

ords

wha

t th

is t

ells

you

abo

ut t

he a

ctua

l per

cent

age

of r

egis

tere

d vo

ters

who

sup

port

incr

ease

d sp

endi

ng

for

edu

cati

on.

Sam

ple

an

swer

:Th

ere

is a

95%

ch

ance

th

atth

e ac

tual

per

cen

tag

e o

f vo

ters

su

pp

ort

ing

incr

ease

d f

eder

al s

pen

din

gfo

r ed

uca

tio

n is

bet

wee

n 5

4% a

nd

62%

.

Hel

pin

g Y

ou

Rem

emb

er

3.T

he f

orm

ula

for

mar

gin

of s

ampl

ing

erro

r m

ay b

e tr

icky

to

rem

embe

r.A

goo

d w

ay t

o st

art

is t

o th

ink

abou

t th

e va

riab

les

that

mu

st b

e in

clu

ded

in t

he

form

ula

.Wh

at a

re t

hes

eva

riab

les,

and

wh

at d

o th

ey r

epre

sen

t? W

hat

is

an e

asy

way

to

rem

embe

r w

hic

h v

aria

ble

goes

in

th

e de

nom

inat

or i

n t

he

form

ula

?S

amp

le a

nsw

er:

pis

th

e p

rob

abili

ty o

fa

cert

ain

res

po

nse

an

d n

is t

he

sam

ple

siz

e.T

he

larg

er t

he

sam

ple

siz

e,th

e sm

alle

r th

e m

arg

in o

f er

ror,

so n

mu

st g

o in

th

e d

eno

min

ato

r si

nce

div

idin

g b

y a

larg

er n

um

ber

giv

es a

sm

alle

r n

um

ber

.Th

e sq

uar

e ro

ot

of

asm

alle

r n

um

ber

is a

sm

alle

r n

um

ber

,an

d t

wic

e th

e sq

uar

e ro

ot

of

asm

alle

r n

um

ber

is a

sm

alle

r n

um

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.

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of

Dis

trib

uti

on

Cu

rves

Gra

phs

of f

requ

ency

dis

trib

uti

ons

can

be

desc

ribe

d as

eit

her

sym

met

ric

or s

kew

ed.

In a

dis

trib

uti

on s

kew

ed t

o th

e ri

ght,

ther

e ar

e a

larg

er n

um

ber

of h

igh

valu

es.T

he

lon

g “t

ail”

exte

nds

to

the

righ

t.

In a

dis

trib

uti

on s

kew

ed t

o th

e le

ft,t

her

e ar

e a

larg

er n

um

ber

of l

ow v

alu

es.

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e “t

ail”

exte

nds

to

the

left

.

For

eac

h o

f th

e fo

llow

ing,

stat

e w

het

her

th

e d

istr

ibu

tion

is

sym

met

ric

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kew

ed.I

f it

is

skew

ed,t

ell

wh

eth

er i

t is

sk

ewed

to

the

righ

t or

to

the

left

.

1.2.

3.

sym

met

ric

skew

ed t

o t

he

left

skew

ed t

o t

he

rig

ht

4.5.

6.

sym

met

ric

sym

met

ric

skew

ed t

o t

he

rig

ht

A v

erti

cal

lin

e ab

ove

the

med

ian

div

ides

th

e ar

ea u

nd

er a

fre

qu

ency

curv

e in

hal

f.

7.W

her

e is

th

e m

edia

n i

n a

sym

met

ric

8.W

her

e is

th

e m

edia

n i

n a

ske

wed

dist

ribu

tion

?In

th

e m

idd

le o

f th

e di

stri

buti

on?

To t

he

left

of

the

mid

dle

ra

ng

e;it

is t

he

sam

e as

th

e m

ean

.if

ske

wed

to

th

e ri

gh

t;to

th

e ri

gh

to

f th

e m

idd

le if

ske

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to t

he

left

.

Sym

met

ric

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to

th

e R

igh

tS

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o t

he

Lef

t

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ian

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ian

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rich

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____

____

____

____

____

____

____

____

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____

____

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____

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12-9

12-9


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. B

C

B

B

C

A

B

A

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An

swer

s

(continued on the next page)

Chapter 12 Assessment Answer Key Form 1 Form 2APage 753 Page 754 Page 755


12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:Sample answer:

{7, 9, 18, 20, 24, 40, 50}

A

C

C

A

A

B

D

C

A

D

D

A

B

C

C

D

B

D

A

C

Sample answer: {7, 10, 17, 24, 26, 28, 28}

A

C

C

D

A

B

A

B

C

Chapter 12 Assessment Answer Key Form 2A (continued) Form 2BPage 756 Page 757 Page 758


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 74; 4

about 9%

No, the opinions of oneclass may not be typicalof all members of their

age group.

47.5%


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Chapter 12 Assessment Answer Key Form 2CPage 759 Page 760


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 77; 3

about 13%

No, library card holdersmay not have opinionsthat are typical of the

community.

47.5%

positively skewed

0.15 in.

0.02

Sample answer: Median; it is

closer to most ofthe values.

630

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�2956�

�290�

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105

Chapter 12 Assessment Answer Key Form 2DPage 761 Page 762


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: about 1275 students

about 348 people

No, drivers may nothave the same opinion

as nondrivers in thetown.

0.05

�23

2,03328

�

4890

positively skewed

$356.20

126,875.81

Sample answer:Mean; the

Maryland taxesare above the

median but theyare below the

mean.

34,560

�25251

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�1334�

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1,66950

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162,162

70; combination; orderdoes not matter.

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An

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Chapter 12 Assessment Answer Key Form 3Page 763 Page 764


Chapter 12 Assessment Answer KeyPage 765, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts of solvingproblems involving finding probability, independent anddependent events, permutations, combinations, mutuallyexclusive and inclusive events, statistical measures, andthe normal distribution

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of solvingproblems involving finding probability, independent anddependent events, permutations, combinations, mutuallyexclusive and inclusive events, statistical measures, andthe normal distribution

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofsolving problems involving finding probability, independentand dependent events, permutations, combinations,mutually exclusive and inclusive events, statisticalmeasures, and the normal distribution

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofsolving problems involving finding probability, independentand dependent events, permutations, combinations,mutually exclusive and inclusive events, statisticalmeasures, and the normal distribution

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy the requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations


1a. Student responses must indicate thatAlma’s solution is correct. Explanationsshould indicate that, since A and Brepresent two independent events andthey are looking for the probability thatboth events occurred, the twoprobabilities should be multiplied.Addition would be required if they werelooking for the probability of either oneof the events to occur.

1b. Sample answer for Steven’s solutionP(A) � P(B) � P(A and B) �

�26� � �

36� � �

16� � �

46� � �

23�: A die is rolled.

Find the probability that a numbergreater than 4 or an even number isrolled.

2a. The student response should indicatethat for grades listed, left to right, fromlowest to highest, a negatively skeweddistribution would include a greaternumber of high scores than low scores.Thus, the student should be happy!

2b. Students should explain that the mean,median, and mode of a normaldistribution are the same, so the meancan be presumed to be

�56 �

298

� � 77, or very close to 77. The

fact that there are three standarddeviations between 77 and 98 (orbetween 56 and 77) means that the

standard deviation is �98 �3

77� � 7

�or �77 �3

56� � 7�. Thus, scores in the

range 77 � 7, or between 70 and 84,would earn a grade of C.

3a. Sample answer: For 6 dinner guests,there would be 8 players including Gregand Jacqui, meaning that there wouldbe 70 different ways to arrange theguests in two teams; students shouldindicate that this is a problem involvingcombinations, rather than permutationssince changing the order in whichplayers are selected for each team wouldnot result in the formation of differentteams.

3b. Students should state that this newcondition would, in fact, change thenumber of arrangements. Taking Gregand Jacqui out of the situation for themoment, the question, for the sampleanswer in part a, would become: In howmany ways can you divide a group of 6people into two groups of 3 people each?The number of ways to do so would be C(6, 3) � 20. Then, since there are twoways to place Greg with one group andJacqui with the other, there are only 20 � 2 � 40 possible arrangements ifGreg and Jacqui cannot be on the sameteam.

An

swer

s

Chapter 12 Assessment Answer Key Page 765, Open-Ended Assessment

Sample Answers

In addition to the scoring rubric found on page A34, the following sample answers may be used as guidance in evaluating open-ended assessment items.


1. false; combination

2. true

3. false; standarddeviation

4. false; discreteprobabilitydistributions

5. false; binomialexperiment

6. true

7. true

8. false; compoundevents

9. false; measures ofcentral tendency

10. false; skeweddistribution

11. Sample answer:Mutually exclusiveevents are eventsthat cannot bothhappen at the sametime.

12. A sample is arandom sampleevery possiblesample of that sizehas an equalchance of beingchosen.

1.

2.

3.

4.

5.

Quiz (Lessons 12–4 and 12–5)

Page 767

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

Quiz (Lessons 12–8 and 12–9)

Page 768

1.

2.

3.

4.

5. about 11%

No, the people surveyedare more likely to prefer

basketball over othersports.

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�

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�

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396

68%

positively skewed

2.17

4.69

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combination; 175

5040

D

Chapter 12 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 12–1 through 12–3) Quiz (Lessons 12–6 and 12–7)

Page 766 Page 767 Page 768


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

1.

2.

3.4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15. about 6%

47.5%

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�

24

�45

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D

D

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An

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Chapter 12 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 769 Page 770


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17. DCBA

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.. ./ /

.

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Chapter 12 Assessment Answer KeyStandardized Test Practice

Page 771 Page 772


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

Sample answer: No;those surveyed aremore likely to listen to a station that airs thetype of music being performed at theconcert.

�54152

�

0.5%

positively skewed

374.88; 356; no mode; 243.67

�173�

�13

�

�16

�

8:5

66

5040

240

Sample answer: n � 2

See students’ answers.

810xy4

1, �3, �11

5, 16, 49, 148, 445

�234353

�

108

96

63

3072, 2304, 1728, 1296

3200, 2560

7

19, 17, 15

22, 28, 34, 40

Chapter 12 Assessment Answer Key Unit 4 TestPage 773 Page 774

An

swer

s

chapter 12 resource masters - ktl math classes . . . . . . . . . . . . . . . . . . . . . .a2–a39...

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