math fundamentals for statistics (math 95)home.miracosta.edu/bpickett/16sp m 95 and 126 and 131/math...

Math 95: Homework Unit 1 –

Math Fundamentals for

Statistics (Math 95)

Homework Unit 1: Logic and Sets

Scott Fallstrom and Brent Pickett “The ‘How’ and ‘Whys’ Guys”


1.1: Fallacies in Reasoning/Argument

Vocabulary and symbols – write out what the following mean:

Logic Premises

Conclusion Valid

Fallacy

Concept questions:

1. What is the difference between the fallacy being related to relevance or being related to assumptions?

Exercises: 2. Determine the type of fallacy “related to relevance” given by these comments.

a. If you love your children, you’ll serve them Ovaltine in the morning. b. Peyton Manning (professional football player) told me to purchase Papa John’s Pizza because it is

better pizza. c. Toyota is the best vehicle company in the world because in 2014, they sold more vehicles than any

other company. d. There has been no evidence showing a conspiracy in the JFK assassination in 1963, so we know

there wasn’t a cover-up. e. You shouldn’t vote for my opponent because he didn’t even finish college – what could he possibly

know about leadership? f. I admit that there could be some issues with email messages through my office, but my crime

package is incredible and will change the way that we respond to future issues with the criminal justice system. Here’s some of the key points: less jail time for minor offenders, more rehabilitation programs, and decreased cost.

g. The governor has worked to decrease taxes paid by business in our state – that means more of the bills are being paid by working class families. It seems that the governor is catering to the big money donors.

h. Having ATD protect your home will provide you security. You don’t know what could happen if you don’t have it.

3. Determine the type of fallacy “related to assumption” given by these comments. a. Jolene can’t trust Lucas because she says that everybody lets her down. Because everybody lets her

down, she can’t trust anyone. b. I’ve been getting 50% on each test so far, and I know I’m due for a big score soon. c. Inception was incredible and so was Interstellar, and both were made by Chris Nolan. I will like

every movie he makes. d. If we buy stuff like this at Costco, then we’ll be out of money & living in a van down by the river. e. America has more people in jail than Australia. Matt Damon (an American) is more likely a

criminal than Hugh Jackman (an Australian). f. When I wear sunglasses and stare into the sun, my eyes hurt. My sunglasses make my eyes hurt. g. You could buy another car, but I thought you said that you loved your family and wanted to keep

them safe? That’s why you need this Kia Soul. h. The last few times I picked the Seahawks to win, they didn’t. I know they are ready for a big win

soon so I’m going to choose them again. i. Paying 39.6% as the highest tax bracket is too much. Years ago, it was only 25%, so it seems that

about 30% would be best.


Wrap-up and look back: 4. Would it be considered false authority to have Tom Brady, a professional football player, tell you to

purchase Nike footballs? 5. Write in words what you learned from this first section. Did you have any questions remaining that

weren’t covered in class? Write them out and bring them back to class.

1.2: Types of reasoning Vocabulary and symbols – write out what the following mean:

Inductive Reasoning Deductive Reasoning

Concept questions: 1. Which type of reasoning takes specific examples and creates a general conclusion?

2. Which type of reasoning takes a general piece of knowledge and applies it to a specific situation?

Exercises: 3. Determine the type of reasoning used in these examples and decide if the conclusion is guaranteed. Statements Conclusion Type of

Reasoning Conclusion

Guaranteed?

a. I ate at McDonald’s and it was good. Then I ate at Wendy’s

and I loved it.

I will enjoy eating at Wendy’s.

Deductive

Inductive

Yes

No

b. The Associative Property of

Multiplication is true. 237237

Deductive

Inductive

Yes

No

c. A vegetarian doesn’t eat any

meat. Paco ate a chicken taco. Paco is not a vegetarian.

Deductive

Inductive

Yes

No

d. Green Lantern was a huge

flop. John Carter was a huge flop.

Sci-fi movies have a good chance to flop, so

Star Wars will flop.

Deductive

Inductive

Yes

No

e. My tummy hurts when I eat

cheese. My tummy hurts when I eat ice cream.

I must be lactose intolerant.

Deductive

Inductive

Yes

No

f. Volunteers help others for no pay. Marissa collects clothing

to donate to the homeless. Marissa is a volunteer.

Deductive

Inductive

Yes

No

g.

The Toyota Prius gets more than 40 mpg highway. The Toyota Corolla gets over 40

mpg highway.

My new Toyota Land Cruiser will get more than 40 mpg highway.

Deductive

Inductive

Yes

No


4. Is this inductive or deductive reasoning? Noel is asked to find the next term in the geometric sequence: 4, 8, 16, … and she comes up with 32. Explain why.

5. Is this inductive or deductive reasoning? Noel is asked to find the next term in the sequence: 4, 8, 16,

… and she comes up with 32. Explain why. 6. Is this inductive or deductive reasoning? Noah ate Thai food and loved it. Then he tried Italian food

and really liked it too. Noah is going to love the Indian food we are going to tonight.

Wrap-up and look back: 7. What is the point in having reasoning (like inductive) that doesn’t guarantee a conclusion? 8. Write in words what you learned from this first section. Did you have any questions remaining that


1.3: Formal Logic – Statements and Quantifiers. Vocabulary and symbols – write out what the following mean:

Propositions Statements Quantifiers

All Some None

Negation Truth Table Equivalent Statements

Concept questions: 1. A sentence like “Are you my mother” is not a statement. Why can there be no truth value for this

sentence? 2. Lisa thinks the negation of “All G are P” would be “Some G are P.” Is this correct? Explain why or

why not. 3. Adam Lambert thinks the negation of “Some G are P” would be “Some G are not P.” Is this correct?

Explain why or why not. 4. Melissa McCarthy thinks the negation of “No G are P” would be “All G are P.” Is this correct? Explain

why or why not. 5. Tina Fey thinks the negation of “No G are P” would be “All G are not P.” Is this correct? Explain why

or why not. 6. Pedro claims that the negation of “Some G are not P” would be “All G are P.” Is this correct? Explain

why or why not. 7. If there were exactly 995 cases of something happening and 999 different ways it could happen, which

quantifier is correct: All, Some, or None? Explain. 8. If there were exactly 0 cases of something happening and 999 different ways it could happen, which

quantifier is correct: All, Some, or None? Explain. 9. If there was exactly 1 case of something happening and 999 different ways it could happen, which

quantifier is correct: All, Some, or None? Explain.


10. If there were exactly 999 cases of something happening and 999 different ways it could happen, which quantifier is correct: All, Some, or None? Explain.

Exercises: 11. Determine whether the sentence is a statement or not, then if it is a statement, find the truth value. Sentence Proposition Truth Value

a. Barack Obama is our current president. Yes No True False

b. 17 is prime. Yes No True False

c. Abe Lincoln was the fifth president. Yes No True False

d. 5 + 8 < 13 Yes No True False

e. 79 > 45 Yes No True False

f. x + 15 = 23 Yes No True False

g. This sentence is false. Yes No True False

h. Scott Fallstrom is 6 feet tall. Yes No True False

i. All Californians are blonde. Yes No True False

j. I’m lovin’ it. Yes No True False

k. Some frogs are green. Yes No True False

l. All women are excellent in math. Yes No True False

m. Some cars are dirty. Yes No True False

n. No MiraCosta student is under 20 years old. Yes No True False

o. No drugs are good for you. Yes No True False

p. All cartoons are not appropriate for kids. Yes No True False

q. Minimum wage is $30 per hour. Yes No True False

r. Some kids behave well. Yes No True False

12. Write the negation of the following statements:

a. All cartoons are not appropriate for kids. b. Some cars are dirty. c. No drugs are good for you. d. Minimum wage is $30 per hour. e. 79 > 45 f. 16 + 30 = 46

g. All Californians are blonde. h. It is 90 degrees outside. i. The grass is purple. j. I am sad. k. Today is a good day. l. Marsha lost the game.

13. Fill out the following truth table:

p q p~ q~ p~~ q~~

a) T F

b) F T


Wrap-up and look back: 14. Can an opinion like “Star Wars is better than Star Trek” be a statement? Explain. 15. Marlesha takes the statement R and creates the negation. Then Bertha negates this new statement and

calls it W. How does W relate back to statement R? 16. How does “All cars are fast” relate to “No cars are not fast”? Are they equivalent? Why or why not. 17. Write in words what you learned from this first section. Did you have any questions remaining that


1.4: Formal Logic – Operators (Truth Tables). Vocabulary and symbols – write out what the following mean:

Compound Statements Conjunction Conditional Condition

Conclusion De Morgan’s Laws qp

qp

qp

Converse Inverse Contrapositive

Concept questions: 1. If only one of the two simple statements in qp is true, does that mean qp is true or false? Why?

2. If only one of the two simple statements in qp is true, does that mean qp is true or false? Why?

3. If only one of the two simple statements in qp is true, does that mean qp is true or false? Why?

4. If qp is true, then what is the truth value of the contrapositive of qp ?

Exercises: 5. Determine the truth values of the statements listed using a truth table.

p q qp p~ q~ qp qp ~ qp ~~ qp ~ qp ~~

a. T F

b. F F

c. T T

d. F T

6. What is the contrapositive of the following (put answer in symbols):

a. pq

b. wr ~

c. Lh ~

d. pq ~~


7. When is the only time that a conditional statement can be false? Explain.

8. Determine the truth values of the statements listed using a truth table.

p q qp p~ q~ qp pq qp ~~ qp ~ pq ~~

a. T F

b. F F

c. T T

d. F T

9. Find an equivalent expression to the given statement (write your answer in English, not symbols).

a. If it is rainy, then we will not go to the game.

b. If my sister has a girl, then we will buy a new car.

c. If the hostages are not released, then we will bomb the city.

d. If it is not sunny, then we will go to a movie.

10. What would it take to prove the following statements false?

a. All rich people own 4 homes.

b. Some turtles eat cheese.

c. Some kids are not well behaved.

d. No divorces are easy.

e. All birds eat seeds.

f. Some chipmunks are not friendly.

g. No math problems are hard.

h. Some history papers are long.

Summary of the negations to use in the next few questions:

Negation of negation: pp ~~

Negation of conjunction: qpqp ~~~

Negation of disjunction: qpqp ~~~

Negation of conditional: qpqp ~~ 11. Use your negation knowledge to negate the following statements (answer as an English sentence):

a. If I am sleepy, then I drive well.

b. All divers use snorkels.

c. Some fast food is healthy.

d. It is cloudy and 5 + 5 = 12.

e. I am sad or you do not wear hats.

f. You are not crazy.

g. Kale is healthy and kale tastes bad.

h. No people like to read.

12. Use your negation knowledge to negate the following statements (answer with symbols):

a. qp ~

b. wp ~

c. qr ~~

d. qp ~

e. q~

f. hr ~~


13. Complete the following parts involving negations and equivalent statements.

a. What is an equivalent way to write qp using the contrapositive?

b. What is the negation of qp using symbols?

c. Take what you found in part (b) and determine the negation using De Morgan’s laws.

d. Since pp ~~ , determine another equivalent statement to qp that is not a conditional

statement.

14. Determine the truth values of the given statements if we know the following: p: 12 + 8 = 20 q: 15 < 9 a. qp b. qp

c. qp d. p~

15. Determine the truth values of the given statements if we know the following: p: 12 + 8 = 30 q: 15 < 9

a. qp b. qp

c. qp d. p~

Wrap-up and look back: 16. What is the point in having reasoning (like inductive) that doesn’t guarantee a conclusion?

17. Write in words what you learned from this first section. Did you have any questions remaining that


1.5: Expanded Conditionals and Bi-Conditionals. Vocabulary and symbols – write out what the following mean:

Bi-conditional Only if Contrapositive

Concept questions: 1. Lucille says that “you are happy if it is sunny” is the same thing as “if it is sunny, then you are happy.”

Is she correct? Explain.

2. Francis says that “you are happy only if it is sunny” is the same thing as “if it is sunny, then you are

happy.” Is he correct? Explain.

3. Justine says that “you are happy only if it is sunny” is the same thing as “if you are happy, then it is

sunny.” Is she correct? Explain.

4. Consider the statement: “A number is even if and only if the number ends in a 0, 2, 4, 6, or 8.” Can you

use this to tell if a number is even? Explain.


Exercises: 5. Write the following symbolically as if-then statements. p: you are angry q: it is rainy r: roses are red it is cold

a. You are angry if it is rainy. b. Roses are red only if it is cold. c. If it is cold, then you are not angry. d. It is rainy only if it is cold. e. It is cold if it is not rainy. f. Roses are not red only if you are not angry. g. If roses are red, then it is not cold.

6. You are shown some cards and given the following rule. Your task is to determine which cards must be

turned over to determine if the rule is being followed. Explain your reasoning. Rule: “If the card has brown on one side, then the other side must have a prime number.”

Card 1 Card 2 Card 3 Card 4 Card 5 Card 6 Card 7 Card 8

5 8 Red Green 7 9 13 Brown 7. Find an equivalent statement to these conditionals. Rewrite with De Morgan’s Laws when possible.

a. If Marissa kisses Mason and Josue sees it, then Josue will break up with Marissa. b. If you have used the American Opportunity Credit for 4 years or you make more than $70,000, then

you don’t get to claim the refundable portion. c. If it is raining, then you can’t go to the movie and you can’t play outside. d. If Hope has amnesia or Stefano tries to kill Marlena, then Bo will have to quit his job.

8. Determine the truth values of these statements if you know that p is false, q is true, and r is true. If you

see any other letter for a statement, it could be true or false… we don’t know. Your answer could be true, false, or need more information. Explain your answer.

Statement Truth Value

a. rw ~ True False Need More Information

b. wp True False Need More Information

c. qw True False Need More Information

d. wqp True False Need More Information

e. prq True False Need More Information

f. rqw ~ True False Need More Information

g. wr ~ True False Need More Information


9. A Theorem in mathematics is a statement that is shown to always be true. Here’s a few examples:

a. A triangle with side lengths of a, b, and c is a right triangle if and only if 222 cba . (Pythagorean Theorem and converse)

a. If you know that 222 17158 , what can you conclude from the theorem?

b. If you know that 222 24237 , what can you conclude from the theorem? c. If you have a triangle that is not a right triangle, what do you know about the sides? d. If you have a triangle that is a right triangle, what do you know about the sides?

b. A number ends in a 0 or a 5 if and only if the number is divisible by 5. a. What can you conclude from the theorem about the number 7,890? b. What can you conclude from the theorem about the number 7,894? c. What can you conclude from the theorem about the number 71,911,345? d. If Shalli is thinking of a number and she tells you it is divisible by 5, what can you conclude

about her number?

c. If x is odd, then 12 x is even.

a. If you know that 12 x is odd, what can you conclude from the theorem?

b. If you know that 12 x is even, what can you conclude from the theorem?

c. Based on the theorem, 217 is odd, so can you conclude 1217 2 is even?

d. Based on the theorem, 978 is even, so can you conclude 1978 2 is even?

d. If 6 is a factor of a number P, then 3 is a factor of the number P. a. If you know that 6 is a factor of M, what else (if anything) can you say about M? b. If you know that 6 is not a factor of M, what else (if anything) can you say about M? c. If you know that 3 is a factor of M, what else (if anything) can you say about M? d. If you know that 3 is not a factor of M, what else (if anything) can you say about M?

Wrap-up and look back: 10. When is a conditional statement false?

11. Why are many math definitions written using a bi-conditional?

12. What is the difference between a conditional and a bi-conditional?



1.6: Basic Arguments – Using Logic. Vocabulary and symbols – write out what the following mean:

Premises Conclusion

Valid Fallacy

Invalid


Concept questions: 1. Can an argument be valid and the conclusion be false?

2. Can an argument be valid and the premises be false?

3. What makes an argument invalid?

4. When checking to see if an argument is valid, what assumptions are made about the premises?

Exercises: 5. Determine if the following arguments are valid. Explain your reasoning. a.

1. If you are a gambler, then you are not financially stable. 2. Sean isn’t a gambler. c. Sean is financially stable.

b.

1. If you have a college degree, then you are not lazy. 2. Marsha is not lazy. c. Marsha has a college degree.

c.

1. All cats shed lots of hair. 2. Shannon sheds lots of hair. c. Shannon is a cat.

d.

1. If you have a speeding ticket, then you can’t apply for the job as a pizza delivery person. 2. Beth is able to apply for the job as a pizza delivery person. c. Beth doesn’t have a speeding ticket.

e.

1. All licensed drivers have insurance. 2. If you have insurance, then you obey the law. 3. You obey the law. c. You are a licensed driver.

f.

1. All forest rangers climb trees. 2. If you are afraid of heights, then you don’t climb trees. 3. Veronica is afraid of heights. c. Veronica is not a forest ranger.

g.

4. All forest rangers climb trees. 5. If you are afraid of heights, then you don’t climb trees. 6. Veronica is not afraid of heights. d. Veronica is a forest ranger.


6. Determine a conclusion that would make the argument valid, if possible. a.

1. If you are a gambler, then you are not financially stable. 2. Hollis is a gambler. c.

b.

1. If a defendant is innocent, then he does not go to jail. 2. Podric is innocent. c.

c.

1. If you listen to speed metal, then you don’t go to heaven. 2. If you are a good person, then you go to heaven. 3. Rachel listens to speed metal. c.

d.

1. If you get an audit, then you didn’t fill out your tax forms correctly. 2. If you win the lottery, then you will get an audit. 3. Tyson won the lottery. c.

e.

1. All doctors are mean. 2. All mean people have good karma. 3. My mother has bad karma. c.

f.

1. If you earn over $14,000 or are allergic to cats, then you need to fill out form W. 2. Karl earns $18,000. 3. Karl is not allergic to cats. c.

g.

1. If you earn over $14,000 or are allergic to cats, then you need to fill out form W. 2. Carleen earns $8,000. 3. Carleen is allergic to cats. c.

h.

1. If you earn over $14,000 or are allergic to cats, then you need to fill out form W. 2. Chase does not need to fill out form W. c.


7. Practice another logic puzzle.

A chess tournament is occurring in the local community school, and the players at all four of the tables are engaged in their fourth game against their prospective opponents.

The players with white pieces are: David, Gerry, Lenny and Terry The players with black pieces are: Don, Mike, Richie and Stephen The scores are 3:0, 2.5:0.5, 2:1, 1.5:1.5 [note: tied games result in a score of 0.5 points for each player]

Lenny is playing at the table to the right of Stephen, who has lost all of his games until now. Gerry is playing against Mike. At least one game at table 1 has resulted in a tie. Richie, who is not in the lead over his opponent, has not been in a tied game. The player who is using the white pieces at table 4 is Terry; however, the current score at table 4 is not 2:1. Don is leading his match after his last three games.

I. What table is Stephen playing at, and what is the score at that table?

A. Table 1, 2.5:1.5

B. Table 1, 3:0

C. Table 2, 3:0

D. Table 2, 2.5:1.5

E. Table 3, 2:1

II. Whose score is highest?

A. Mike B. Stephen C. Richie D. David E. Lenny

III. Which player has black pieces and is tied?

A. Mike B. David C. Richie D. Don E. Terry

IV. Who is the winning player at table 4?

A. Don B. Terry C. David D. Gerry E. Richie

8. Determine whether there is a problem with the person’s thinking. Be able to explain your reasoning.

a. John’s mom told him “If you get home after 10pm, then I will take your cell phone.” John got home at 10:30pm and his mom didn’t take his phone. He was really ticked off because he said that she lied to him. Did she?

b. Louis read the IRS guideline: “If your income on line 37 is more than $100,000 and you are filing as single, then you must fill out form J.” His income is $75,000 and he is married filing jointly, so he didn’t fill out form J. He was frustrated when he was audited for not filling out his taxes properly. Does his complaint hold water?

Wrap-up and look back: 9. Write in words what you learned from this first section. Did you have any questions remaining that



1.7: Arguments – Using Venn Diagrams. Vocabulary and symbols – write out what the following mean:

Venn Diagrams Sound Argument Not Sound Argument

Concept questions: 1. In a Venn diagram where the 2 circles overlap, how many different regions are there?

2. In a Venn diagram where the 2 circles are not overlapping at all, how many different regions are there?

3. In a Venn diagram where one circle is completely inside another, how many different regions are there?

4.

Exercises: 5. Based on the Venn diagram, determine in

words what type of object each letter represents.

6. Based on the Venn diagram, determine in words

what type of object each letter represents. 7. Label a Venn diagram that represents “Some

cats are bald.” Then write letters in the appropriate places in your diagram.

I. “X” represents a furry dog II. “Y” represents a furry cat.

III. “Z” represents a bald old man. IV. “M” represents a hairless cat.

Purses things that are brown

A B C D

Cars Fast objects

A B C D


8. Based on the Venn diagram, determine in words what type of object each letter represents.

9. Based on the Venn diagram, determine in


10. Label a Venn diagram that represents “All math

students are hard workers.” Then write letters in the appropriate places in your diagram.

I. “X” represents a lazy cat. II. “Y” represents a math student.

III. “Z” represents a lazy jeweler. IV. “M” represents a hard working plumber. 11. Label a Venn diagram that represents “If you

kiss frogs, then you are a princess.” Then write letters in the appropriate places in your diagram.

I. “X” represents a math teacher who doesn’t kiss frogs.

II. “Y” represents a frog kissing princess. III. “Z” represents a princess who never kisses

frogs. IV. “M” represents a hard working plumber who

kisses frogs.

teachers

women

A B C

boys

Objects that smell bad

A B C






14. Label a Venn diagram that represents “No

Democrats own guns.” Then write letters in the appropriate places in your diagram.

I. “X” represents a gun owning Libertarian. II. “Y” represents a Republican who owns no

guns. III. “Z” represents a Republican who owns a

gun. IV. “M” represents a Democrat. 15. Label a Venn diagram that represents “No

dolphin can fly.” Then write letters in the appropriate places in your diagram.

I. “X” represents a dog. II. “Y” represents a flying squirrel.

III. “Z” represents a dolphin. IV. “M” represents an airplane.

things that are brown

lettuce

A B C

Things painted red

boys

A B C


16. Determine if the argument is valid using a Venn diagram. Remember, if there is any way that the conclusion doesn’t follow, then the argument is invalid! Also, determine if this argument is sound.

I. 1. Some cats climb trees. 2. Jonathan climbs a tree. c. Jonathan is a cat.

II.

1. No fast food is tasty. 2. Salads are not tasty. c. Salads are fast food.

III.

1. If you are guilty, then you need an attorney.

2. Sam doesn’t need an attorney.

3. If you commit a crime, then you are guilty.

c. Sam didn’t commit a crime.

Wrap-up and look back: 17. What is the difference between a sound argument and a valid argument? 18. Can an argument be valid and still not sound? Explain.

19. Samantha thinks that “if you are happy, then you will go running” and “John goes running” means that

John is happy. She explains that if he wasn’t happy, he wouldn’t go running. Does this make sense? Explain why or show where Samantha has made her mistake.

20. Write in words what you learned from this first section. Did you have any questions remaining that weren’t covered in class? Write them out and bring them back to class.

1.8: Learning Deductive Reasoning through Games. Vocabulary and symbols – write out what the following mean:

Minesweeper Mastermind Are you the one?

Concept questions: 1. Is there ever a time where you may have to “guess” in Minesweeper? Explain such a time.

2. In Minesweeper, if you click on a cell and uncover the number 8, what information do you know and

how helpful is it?

3. In Minesweeper, if you click on a cell and uncover a number, could you ever uncover the number 9?

Explain why or why not.

4. In Mastermind, if you guess R-O-Y-G and get 4 white pegs as clues, does this help you? Explain.

5. In Mastermind, if you guess R-R-G-G and get 3 white pegs as clues, does this help you? How?


Exercises:

6. Based on the screenshot from Minesweeper, determine which letters are on “bombs” and which letters are on “numbers” and which letters you don’t know yet. a.

b.

c.

d.

e. Find and describe the error in this person’s puzzle – one of the flags is not correctly marking a bomb.

A D B E C F G H I J K L M N O P Q R S T U

A B C D E F G H I

A B C D E F G H I J K L

A B C D E F G H I J K


f. Three flags are incorrect in this person’s

puzzle. Which ones and how do you know?

7. Based on the screenshot from Minesweeper, determine which boxes are on “bombs,” which boxes are on “numbers,” and which boxes you don’t know yet.

8. In Minesweeper, each game starts completely blank. What percent of the squares are bombs in each

game? a. Beginner b. Intermediate c. Expert


9. Determine the solution for these MasterMind puzzles (there are only 2 locations for these). a.

Guess Location 1 Location 2 Clues 1 R R BL 2 B B BL 3 G R WH

CODE R G Y B

R G Y B

b.

Guess Location 1 Location 2 Clues 1 R B 2 B Y WH 3 G G BL

CODE R G Y B

R G Y B

c.

Guess Location 1 Location 2 Clues 1 R R 2 B Y WH-WH

CODE R G Y B

R G Y B

d.

Guess Location 1 Location 2 Clues 1 R R WH 2 B Y BL 3 G Y

CODE R G Y B

R G Y B

e. Is it possible to get clues of BL-WH when playing with only two locations? Explain.

10. Did you create a strategy for the MasterMind puzzle? What is your strategy to getting the code correct?

How many steps/guesses will it take for you in a worst-case scenario?


11. Determine the solution for these MasterMind puzzles (there are only 3 locations for these). a.

Guess Location 1 Location 2 Location 3 Clues 1 Y G B BL 2 G B R

CODE R G Y B

R G Y B

R G Y B

b.

Guess Location 1 Location 2 Location 3 Clues 1 Y G B WH-WH-WH 2 G B R BL-BL

CODE R G Y B

R G Y B

R G Y B

c.

Guess Location 1 Location 2 Location 3 Clues 1 Y G B BL-BL 2 G B R WH-WH-WH

CODE R G Y B

R G Y B

R G Y B

d.

Guess Location 1 Location 2 Location 3 Clues 1 R R B BL-BL 2 G B R WH-WH 3 Y Y B BL-BL

CODE R G Y B

R G Y B

R G Y B

e.

Guess Location 1 Location 2 Location 3 Clues 1 R R R 2 G B R WH-WH 3 Y Y B WH-WH

CODE R G Y B

R G Y B

R G Y B

f.

Guess Location 1 Location 2 Location 3 Clues 1 R R B BL-BL 2 G B R WH-WH 3 Y Y B BL-BL

CODE R G Y B

R G Y B

R G Y B

g. Could one of the clues in this version be BL-BL-WH? Explain.


12. Determine the solution for these MasterMind puzzles (full 4 locations for these, but only 4 colors).

a. Guess Location 1 Location 2 Location 3 Location 4 Clues

1 R B B G WH-WH

2 B Y B Y WH-WH

3 B Y B R BL-WH

CODE R G Y B

R G Y B

R G Y B

R G Y B

b. Guess Location 1 Location 2 Location 3 Location 4 Clues

1 R R G G BL-BL

2 Y Y B B BL-BL

3 B R G R BL

CODE R G Y B

R G Y B

R G Y B

R G Y B

c. Guess Location 1 Location 2 Location 3 Location 4 Clues

1 R Y B Y BL-WH-WH-WH

2 Y B R Y BL-BL-WH-WH

CODE R G Y B

R G Y B

R G Y B

R G Y B

d. Guess Location 1 Location 2 Location 3 Location 4 Clues

1 R B R B BL-WH

2 G Y G Y WH-WH

3 B R G R WH-WH

4 R B R G BL-BL

CODE R G Y B

R G Y B

R G Y B

R G Y B

e. Could one of the clues in this version be WH-WH-WH-WH if the answer had 2 of the same color

and your guess had 2 of that color? Explain.

f. Could one of the clues in this version be WH-WH-WH-WH if the answer had 3 of the same color color and your guess had 3 of that color? Explain.

g. Could one of the clues in this version be BL-BL-BL-WH? Explain.


13. Determine the solution for these MasterMind puzzles (4 locations and 6 colors – the full game).

a.

Guess Location 1 Location 2 Location 3 Location 4 Clues 1 B G G G BL 2 P R Y B BL-WH-WH-WH 3 O Y R G WH-WH 4 G Y O R WH-WH

CODE R O Y G B P

R O Y G B P

R O Y G B P

R O Y G B P

b.

Guess Location 1 Location 2 Location 3 Location 4 Clues 1 B G G G 2 P O O O 3 R Y R Y WH-WH-BL-BL 4 G Y O R BL-BL 5 R O O O BL

CODE R O Y G B P

R O Y G B P

R O Y G B P

R O Y G B P

c.

Guess Location 1 Location 2 Location 3 Location 4 Clues 1 B G R B 2 O P P G BL-WH 3 O Y P G WH-WH-WH 4 G G Y Y BL-WH 5 R R R O WH

CODE R O Y G B P

R O Y G B P

R O Y G B P

R O Y G B P

14. Do the 6 color options make the game harder? Why? 15. Would it be easier to guess the code if you didn’t allow repeats (each color in the answer is used

exactly once)? Explain.

16. Would it be easier to guess the code if you knew that the code did have repeats (at least one color in the answer is used more than once)? Explain.

17. Would it be easier to guess the code if you knew that the code did have multiple repeats (at least one

color in the answer is used more than twice)? Explain.

18. Would it be easier to guess the code if you knew that the code was only one color? Explain.

19. Would it be easier to guess the code if you knew that the code used exactly two colors? Explain.


20. Are you the one: Season 3! (We would do season 2, but they had 2 girls as a perfect match for 1 guy, meaning that one girl may not be selected. We’ll just do standard pairings for our course)

Week 1 2 3 4 5 6 7 8 9 Amanda Mike Mike Austin Alec Tyler Chuck Hunter Chuck Nelson Britni Hunter Zak Hunter Nelson Mike Hunter Devin Hunter Devin Chelsea Connor Alec Connor Connor Connor Connor Connor Connor Connor Chey Nelson Nelson Tyler Zak Devin Austin Tyler Tyler Austin Hannah Chuck Chuck Zak Devin Austin Tyler Zak Zak Zak Kayla Zak Connor Mike Mike Zak Zak Austin Nelson Hunter Kiki Austin Austin Chuck Chuck Chuck Nelson Nelson Austin Mike Melanie Devin Devin Nelson Tyler Hunter Mike Chuck Mike Tyler Rashida Tyler Tyler Devin Hunter Nelson Devin Alec Devin Chuck Stacey Alec Hunter Alec Austin Alec Alec Mike Alec Alec # Correct

2 0 3 2 2 3 3 3 2

Truth Booth Information: Week 1 2 3 4 5 6 7 8 9 10 Couple Kiki &

Hunter Kiki &

Devin

Kiki &

Zak

Britni &

Chuck

Chelsey &

Connor

Kiki &

Chuck

Melanie & Alec

Kiki & Nelson

Britni &

Hunter

Kayla &

Zak Result Not Not Not Not YES Not Not Not Not YES Working by yourself or in small groups, circle the perfect matches in the grid below. You may cross out those that are not perfect matches. There may be more than one answer – if there is, find at least two that would work. Amanda Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Britni Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Chelsea Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Chey Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Hannah Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Kayla Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Kiki Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Melanie Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Rashida Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak Stacey Alec Austin Chuck Connor Devin Hunter Mike Nelson Tyler Zak


Wrap-up and look back: 21. Explain why this game of Minesweeper comes down to a

guess, and no amount of reasoning can prevent that. 22. Why is the following picture such a

great clue? Can you use it to quickly find a few bombs? How many boxes must be bombs based on this picture?

23. Mark the boxes that MUST be bombs.


24. What information does the largest number on screen tell you? Explain.

25. You can play MasterMind with words, instead of “code letters.” See if you can expand your knowledge to the new type. Each guess must be an actual English word, but the clues are the same as before! a.

Guess Location 1 Location 2 Location 3 Location 4 Clues 1 S L A T BL 2 F L A G 3 E V I L BL-BL 4 V O W S 5 G L A D WH

b.

Guess Location 1 Location 2 Location 3 Location 4 Clues 1 L O P E BL 2 P E A R 3 S A I L 4 B R E D BL-BL 5 N E A R WH



1.9: Sets – The Basis for Counting. Vocabulary and symbols – write out what the following mean:

Sets Set Notation Elements Subset Complement Universe

Empty Set Equal Sets Cardinal Number A6 A6

DE

ED

B U An


Concept questions: 1. Can you have a set that has a negative number of elements?

2. Can you have a set that has no elements?

3. Can you have a set that has 5 in it twice?

4. Does the order of the elements in a set make a difference – is ,6 ,1 the same as ,1 ,6 ?

5. If ED , does that mean ED ? Explain.

6. If ED , does that mean ED ? Explain.

7. If ED , does that mean DE is false? Explain.

8. Why is E for any set E? Explain.

9. When we say BA , what does that mean?

10. If B3 is false, what do we know about the number 3? Explain.

Exercises: 11. Determine whether these statements are true for BA , ,99 ,5 and , ,12 ,66 ,23 B . Statement Truth value Statement Truth value

a. A66 True False b. B66 True False

c. A99 True False d. B8 True False

e. B True False f. BA True False

g. 8 True False h. AB True False

i. 8 True False j. B True False

12. Determine whether these statements are true for BA ,99 ,5 , , ,12 ,66 ,23 B , and 23C .

Statement Truth value Statement Truth value

a. B23 True False b. B23 True False

c. B True False d. 23 ,66B True False

e. AC True False f. BC True False

g. CB True False h. AB True False

i. AB True False j. BC True False


13. Determine whether these statements are true for ,6 ,1A and AB , ,23 ,6 ,1 . Statement Truth value Statement Truth value

a. B True False b. 6 True False

c. B True False d. A True False

e. True False f. A True False

g. BA True False h. BA True False

i. True False j. True False

14. Determine whether these statements are true for 6 ,23A , 23 ,6 ,1B , and 1 ,23 ,6C . Statement Truth value Statement Truth value

a. CA True False b. AC True False

c. CA True False d. BC True False

e. CB True False f. CB True False

15. Determine whether these statements are true for 6 ,4 ,3 ,2 ,1U 6 ,1A , and 6 ,2 ,1B . Statement Truth value Statement Truth value

a. B5 True False b. B3 True False

c. A6 True False d. A3 True False

e. U5 True False f. U True False

g. BA True False h. UA True False

i. BA True False j. AB True False

16. Determine the following values for 6 ,4 ,3 ,2 ,1U 6 ,1A , and 6 ,2 ,1B .

a. Un

b. Bn

c. An

d. An

e. Bn

An

f. Un

An

g. Bn

h. Un

i. An

Bn

j. Un

An


17. Try a few more involving cardinality.

a. M = {x | x is a letter in the English alphabet}. Mn =

b. H = {x | x is a letter in the Spanish alphabet}. Hn =

c. S = {x | x is a state in the United States beginning with A}. Sn =

d. W = {x | x is a state in the United States beginning with C}. Wn =

e. F = {x | x is a representative in the United States House of Representatives}. Fn =

f. If 0Pn , what does that tell you about set P?

g. If FnJn , can we conclude that FJ ? Explain.

h. If FJ , can we conclude that FnJn ? Explain.

Wrap-up and look back: 18. How are the symbols and related? Explain. 19. How is the symbol = used differently in these: AnPn and P = A? Explain.

20. Is it possible for an object to be an element of A and a subset of A at the same time? Explain.



1.10: Sets – Basic Operators and Venn Diagrams. Vocabulary and symbols – write out what the following mean:

Intersection Disjoint Union

BAnBnAnBAn

Concept questions: 1. For any two sets, is this statement true: BAnBnAnBAn ? Explain.

2. For any two sets, is this statement true: BnAnBAn ? Explain.

3. For any two disjoint sets, is this statement true: BnAnBAn ? Explain.

4. Is it possible that 0BAn ? Explain.

5. Is it possible for 0BAn ? Explain.


6. Is it possible that 0BAn and 3BAn ? Explain.

7. Is it possible that 3BAn and 0BAn ? Explain.

Exercises: 8. Shade the appropriate regions for each type of graph.

a. Shade B . b. Shade D . c. Shade B. d. Shade U

9. Create a Venn diagram that would represent disjoint sets P and W. 10. Shade the appropriate regions for each type of graph. a. Shade BA . b. Shade AB . c. Shade BA . d. Shade BA . 11. Shade the appropriate regions for each type of graph. a. Shade BA . b. Shade AB . c. Shade BA . d. Shade BA .

12. Shade the appropriate regions for each type of graph. a. Shade BA . b. Shade BA . c. Shade BA .

U B D

U D B

U D B

U A B

U B A

U A B

U A B

U B A

U A B

U A B

U B A

U A B


13. Shade the appropriate regions for each type of graph. a. Shade BA . b. Shade BA . c. Shade BA . 14. Describe what region the lower-case letter is in, for each Venn diagram.

a. b. c. d. 15. For these questions, determine the missing piece and fill out the rest of the Venn diagram.

a. If 600Un , 177An , 29 BAn , and

204Bn , find BAn .

b. If 66Un , 12An , 23 BAn , and 11Bn ,

find BAn .

c. If 550Un , 400An , 400 BAn , and

300Bn , find BAn .

d. If 300Un , 121An , 31 BAn , and 101Bn ,

find BAn .

e. If 975Un , 284An , 800 BAn , and

516Bn , find BAn .

f. If 550Un , 300An , 500 BAn , and

200Bn , find BAn .

g. If 247Un , 124An , 247 BAn , and

123Bn , find BAn .

U A B

U B A

U A B

U P W

U W P

U W P

a b c

a b c

a b c d

U B A


16. There are some clues given for a 3-circle Venn diagram. Use the clues to figure out the rest of the Venn diagram, as well as answering the other questions.

a. 217Un , 70An , and 42Bn .

Find the following: Cn = _________

BAn = __________

CBn = __________

CBAn = __________

b. 483Un , 120An , and 221Bn . Find the following: Cn = _________

BAn = __________

CBn = __________

CBAn = __________

c. 983Un , 120An . Find the following: Bn = _________

CAn = __________

BAn = __________

CBn = __________

U

B A

C

36 15

8 3 23

B A

C

U

22 11 49 87 23

U

B A

C

9 7 104 4 53 66


17. We know that BAnBnAnBAn is true, but how about more sets. What is the

relationship with ?CBAn In this Venn diagram, all portions inside of CBA have been individually labeled and we can identify the pieces. The first two have been done for you. a. CBAn = a + b + c + d + e + f + g

b. An = a + b + d + e

c. Bn =

d. Cn =

e. BAn =

f. CBn =

g. CAn =

h. CBAn =

i. If we said CnBnAnCBAn , have we double counted any of the regions?

j. If we adjusted to be CAnCBnBAnCnBnAnCBAn , does this

work, or is any region included too often or not enough.

k. Fix the formula using only the portions: An , Bn , Cn , BAn , CBn , CAn , and

CBAn . It should be: CBAn =

18. These types of equations are often referred to as inclusion-exclusion. Expand this idea to the next level… DCBAn =

19. If A = {3, 8, 12} and B = {8, 12, 15}, determine:

a. BA b. BA

20. If A = {3, 18, 12} and B = {8, 2, 15}, determine: a. BA b. BA

21. If A = {3, 18, 12}, B = {8, 2, 15}, and C = {2, 11, 12, 18}, determine: a. BA b. BC c. CBA

d. BA

e. BC f. CBA

g. CAB

h. CB

U

a b c d e f g


Wrap-up and look back:

22. Do De Morgan’s Laws apply with sets? That is, does BA = BA in the same way that qpqp ~~~ ? Explain why.

23. Is this true or false: UnAnAn ? Explain.

24. Is it possible that UnBAn ? Explain with a Venn diagram.

25. Is it possible that BAnBAn ? Explain with a Venn diagram.


1.11: Using Sets to Solve Problems. Vocabulary and symbols – write out what the following mean:

None

Concept questions: 1. In a survey with 300 participants, what is the largest number that could be blank?

2. If there are 30 people checking one box and 50 people checking another, what is the maximum and

minimum number who checked both boxes if…

a. There are100 surveys given out.

b. There are 70 surveys given out.

c. There are 60 surveys given out.

d. There are 50 surveys given out.

3. If there are 30 people checking one box and 50 people checking another, what is the maximum and

minimum number who checked no boxes if…

a. There are100 surveys given out.

b. There are 70 surveys given out.

c. There are 60 surveys given out.

d. There are 50 surveys given out.


Exercises: 4. Fill in a 2-circle Venn diagram with the numbers stated, and answer any questions.

a. 42 people showed up to the booth at the farmer’s market where white eggs and brown eggs are sold. 23 purchased white eggs and 9 bought brown eggs, with 5 people buying both. How many people didn’t buy any eggs?

b. 317 people were asked about which apps they had installed on their smartphones. 95 people had “Clash of Clans” and 102 had “Candy Crush.” 180 people had “Clash of Clans” or “Candy Crush.” How many people had both “Clash of Clans” and “Candy Crush”? How many people had neither of these apps?

c. 250 surveys were passed out to gamers who were asked about which system they would love to get as a gift: 76 said PS4 (only), 160 said Xbox One, and there were 98 who wanted to get both. How many wanted only Xbox One? How many wanted neither?

Wrap-up and look back: 5. For this information, 16An and 11Bn , why is it possible that the number in each region can

change for the same information?


1.12: Basic Counting Techniques Vocabulary and symbols – write out what the following mean: Fundamental Counting

Principle One-to-one Correspondence

Factorial n! Permutations

! !

rn

nPrn

Concept questions: 1. For each logic statement, there are 2 options (true or false). In the truth table, that creates 2 rows. But if

there are two statements, then each has 2 options (true or false). How many rows are needed if we have… a. 2 statements b. 3 statements c. 4 statements

d. 5 statements e. 6 statements f. n statements

2. How can you tell whether to use permutations or factorial?

U


3. What’s the difference between using factorial and the multiplication principle?

Exercises: 4. Determine the number of options based on the information given.

a. How many different outfits can be made for a math teacher who has 5 pairs of slacks, 10 shirts, and 6 pairs of shoes?

b. How many ways can you eat a dinner option if you have to pick a salad, a main entrée, and a

dessert… and there are 3 salad options, 5 main entrées, and 2 dessert choices?

c. A noodle company serves 14 types of noodles, 6 protein options, and 11 sauces. If you choose a different order every day, how many days can you eat before you repeat a dish?

d. At McDonald’s, you can order a burger with the following choices: Ketchup (none/extra/regular); Mustard (none/extra/regular); Onions (dehydrated/regular/none); Pickles (none/regular/extra); Cheese (none/2 slices/1 slice); Mayo (none/extra/regular); Lettuce (leaf/shredded/none); and finally Tomato (none/1 slice). How many different ways can you order a burger?

5. There are 6 men and 5 women who will be positioned for a talk show. If the seating must alternate

male-female, how many different ways can the group be seated? 6. Consider the MasterMind game we played earlier. There are 6 different colors and 4 spaces for the code

in the standard game. Which of the following provides more options for codes: a. Code can use only 3 colors (repeating allowed) b. Code can use 4 colors (repeating not allowed)

7. Consider the MasterMind game we played earlier. There are 6 different colors and 4 spaces for the code

in the standard game. Which of the following provides more options for codes: a. Code is extended to 5 spaces with 6 colors (repeating not allowed) b. Code is extended to 8 colors in the 4 spaces (repeating allowed) c. Code is extended to 5 spaces with 6 colors (repeating allowed)

8. Consider the MasterMind game we played earlier. How many different codes are allowed for the

following game versions: a. MasterMind for Kids: 3 code spaces, 6 colors. (repeating allowed) b. MasterMind (standard): 4 code spaces, 6 colors. (repeating allowed) c. MasterMind Ultimate: 5 code spaces, 8 colors. (repeating allowed)

9. Practice using the calculator on these and find the value.

a. 11! b. 0!

c. 2! d. 9!


10. Find the exact value of the following using FLOF:

a. ! 299

! 300 b.

! 499,36

! 500,36 c.

! 398

! 400

11. How many different ways can you arrange the 10 bombs in a game of Minesweeper (Beginners – 100 spaces)?

12. Determine the number of ways to do what is described.

a. Determine how many different orders are possible for a beauty pageant with 6 time-slots and 6 contestants.

b. Determine how many ways we can pair up 11 Bridezillas with 11 amazing gowns. c. Determine how many different ways the men and women can pair up in the MTV show “Are you

the one?” if it was expanded to 12 men and 12 women. d. Determine how many 1-to-1 correspondences are possible between {A, B, C, D} and {1, 2, 3, 4} if

A must map to an even number. e. Determine how many 1-to-1 correspondences are possible between {A, B, C, D, E} and {1, 2, 3, 4,

5} if A must map to an odd number, and D must map to 4. f. Determine how many 1-to-1 correspondences are possible between {A, B, C, D} and {1, 2, 3, 4} if

A must map to an even number.

13. Determine the number of ways that the objects can be arranged in order. a. There are 6 different books on a shelf. b. There are 9 different shirts on hangers in a closet. c. There are 7 students in a spelling bee. d. There are 2 healthy choices on the menu.

14. The serial number on a dollar bill consists of a letter followed by 8 digits and then another letter. How

many different serial numbers are possible if: a. letters and digits can’t be repeated. b. letters and digits can be repeated. c. the letters are vowels and the digits can be repeated. d. The letters are non-repeated consonants, and the digits can be repeated.

15. A class of art students must present their portfolios at the end of the semester. If there are 9 students in the class (6 girls, 3 boys), how many presentation orders are possible if: a. the names are put in a hat and drawn at random? b. all the girls must go before any boy can go? c. all the boys must go together? d. the names are put in alphabetical order?

16. Use your calculator to determine the following values: a. 47 P

b. 411 P

c. 1214 P

d. 1718 P

e. 497 P

f. 310 P


17. Find the results by writing it out in the formula then checking with the calculator. Interpret what each means. a. 06 P

b. 66 P

c. 26 P

d. 6!

18. A Social Security Number (SSN) is 3 digits, followed by 2 digits, followed by 4 digits. How many different SSNs are possible if… a. the digits cannot repeated? b. the digits can be repeated?

19. MiraCosta has phased out using the SSN as a student ID number, but are looking for a new system. The

administration has considered this plan: use 2 of the letters from the set {D, O, L, P, H, I, N, S},

followed by 6 digits. How many ID numbers are possible if …

a. the letters and digits cannot be repeated?

b. the letters and digits can be repeated?

c. the letters can be repeated but the digits cannot?

d. the letters cannot be repeated but the digits can?

20. Determine the number of ways to select the following.

a. Ranking the top 3 books, in order, out of a book of the month club (12 different books).

b. Ranking the top 8 contestants, in order, out of a group of 51 competing for Miss Teen USA.

c. Ranking the top 2 students, in order, in a scholarship competition with 100 different students.

d. Ranking the top 4 burritos, in order, in a taste-test challenge with 400 burritos.

e. Ranking the top 5 t-shirts, in order, from your closet that has 36 t-shirts in it.

21. In the game of CLUE, one person, one room, and one weapon are selected and hidden from view. How

many different 3-card groups can be made if we know there are:

6 people (Col. Mustard, Professor Plum, Mr. Green, Mrs. Peacock, Miss Scarlet, Mrs. White)

6 weapons (Knife, Candlestick, Revolver, Rope, Lead Pipe, Wrench)

9 rooms (Hall, Lounge, Dining Room, Kitchen, Ballroom, Conservatory, Study, Billiard Room,

Library)

22. In the game CLUE – Master Detective, one person, one room, and one weapon are selected and hidden

from view. How many different 3-card groups can be made if we know there are 10 different people, 8

different weapons, and 12 different rooms?


23. In CLUE, you make a guess about who you think did it and then if anyone has a card that matches your guess, they are required to pass it to you (so only you can see). If you use your reasoning, skills, you can often pick up other clues. You and your friends are sitting so that the order of turns is: You, LeBron James, Michelle Wie, and Lionel Messi. If someone who is in front of you in the order passes a card, then you don’t have to. You have the cards: Colonel Mustard, Miss Scarlet, Lead Pipe, Wrench, and Ball Room.

Person

Guessing Guess Made Any Cards Passed This

means?

a. You Colonel Mustard with the knife in the Library. None.

b. LeBron Colonel Mustard with the gun in the Ballroom. Michelle passes a card.

c. Michelle Professor Plum with the wrench in the Ballroom. You pass a card.

d. Lionel Professor Plum with the gun in the Library. LeBron passes a card.

e. You Mr. Green with the lead pipe in the Library. LeBron passes a card.

f. LeBron Mr. Green with the wrench in the Conservatory. Lionel passes a card.

g. Michelle Miss Peacock with the wrench in the Ballroom. Lionel passes a card.

h. Lionel Colonel Mustard with the rope in the Conservatory. You pass a card

i. You (win the game by what guess?)

24. There are many people who play CLUE where they only pay attention when it is their turn. Why is this

a bad strategy? Explain. Wrap-up and look back: 25. Will your calculator work for all types of factorial? If not, what is the limit?

26. Explain which is easier to type into the calculator: ! 47

! 50 or 350 P ?



1.13: Advanced Counting Techniques Vocabulary and symbols – write out what the following mean:

Permutation of like objects Combination

! !

!

! rnr

n

r

PC rn

rn


Concept questions 1. How can you immediately tell that the problem involves permutation of like objects?

2. What’s the difference between permutations and combinations? Explain.

3. For nearly every situation, explain which is true and why:

a. there are more permutations than combinations b. there are more combinations than permutations

Exercises: 4. Determine the number of ways to arrange the objects listed:

a. How many ways can you rank the top 3 bagels, in order, from a set of 13 bagels? b. There are 5 different colored pegs and you are making a 3 color code where you can repeat options.

How many “codes” can be made? c. How many ways are there to rearrange the letters in SPAN? (Are any of them real words?) d. How many ways are there to rearrange the letters in TOOTH?

5. How many unique ways can we rearrange the following word or words?

a. ALABAMA b. HOBBITS c. LAS VEGAS d. MISSION

e. STAR WARS f. STAR TREK g. BILBO BAGGINS h. HARRY POTTER

6. A dog race is held between 4 Pugs, 3 Chihuahuas, 5 Yorkshire Terriers, and 3 Scottish Terriers (Scottie

Dogs ). How many different ways can they finish the race, if only the breed is considered? 7. You are arranging doughnuts on a table and the order makes a difference. How many ways can you

arrange 6 glazed, 4 chocolate, 3 apple fritters, and 2 raspberry filled?

8. How many different 2-card hands can be drawn if: a. It is a standard 52-card deck and repetition is allowed. b. Both cards are dealt at the same time (no repetition).

9. The previous problem makes many people think of the game Blackjack, or 21. There are 4 Aces, 4

Kings, 4 Queens, 4 Jacks, and 4 Tens in a standard deck of cards. If we count each King, Queen, Jack, or Ten as “10” and an Ace as “11,” how many different ways can we get a sum of 21 with a single 2-card hand?

10. In more realistic problems, determine the number of ways to rearrange the items.

a. How many ways can you arrange 5 books on a shelf if 2 books are identical? b. How many ways can you arrange 20 DVDs if there are 5 identical copies of Pulp Fiction, 4

identical copies of Avatar, and 3 identical copies of Titanic?


11. Scott once owned a CD-Jukebox, which was a CD player that held 50 CD’s. Yes, he was that cool! He put in 10 different CD’s, and left 40 spaces empty. How many different ways were there for him to do this?

12. From the previous problem, Scott (the Pandora of his time), put in 10 more CD’s and all 10 happen to

be ABBA – Greatest Hits GOLD! So now there are 20 CD’s in the player, but only 11 different titles. How many different ways were there for him to do this?

13. Compute the following using the combination button on your calculator.

a. 311C b. 352 C c. 721C

14. There are 4 elements in a set {A, B, C, D}. Write out the subsets as you find how many subsets can be

formed with:

a. 0 elements?

b. 1 element?

c. 2 elements?

d. 3 elements?

e. 4 elements?

15. There are 7 elements in a set. How many subsets can be formed with:

f. 0 elements?

g. 1 element?

h. 2 elements?

i. 3 elements?

j. 4 elements?

k. 5 elements?

l. 6 elements?

m. 7 elements?

16. Consider the set of letters {e, m, p, t, y}.

a. How many permutations of 2 letters can be made? b. How many combinations? c. List all the combinations.

17. A baseball league has 14 teams. If every team must play each other team once in the first round of

league play, how many games must be scheduled? 18. This same baseball league (from the previous problem) completes the season. How many different end

of the season rankings are there for first, second, and third place? (assume no ties).

19. In a group of 39 people, each person is to introduce themselves to each other person, and shake hands. How many handshakes take place?

20. Two hundred people buy 1 raffle ticket each. 3 winning tickets will be drawn. How many different

ways can the prizes be awarded if a. the first prize is $100, the second is $50, and the third is $20? b. if each prize is $50?


21. Opal has 32 DVDs, but can only take 4 on her long drive. a. Does the order of selection matter? b. How many ways can she select 4 DVDs? c. How many different ways could she watch the 4 DVDs if she doesn’t repeat movies?

22. A committee of 4 is to be chosen from a group of 15 people. How many committees are possible if:

a. there is no distinction between the members? b. one person is the chair, the rest are general members? c. one is the chair, one is secretary, one is treasurer, and one is the janitor?

23. A group of 8 women and 6 men must select a 4 person committee. How many committees are possible

if it must consist of: a. 2 women and 2 men? b. any mixture of men and women? c. a majority of women?

d. a majority of men? e. no men? f. at least one man?

24. A group of 6 seniors, 5 juniors, 4 sophomores and 3 freshmen must select a committee of 4. How many

committees are possible if it must consist of: a. one person from each class? b. any mixture of the classes?

c. exactly 2 seniors? d. more than 2 freshmen?

25. In Arizona there is a 5-35 lottery, meaning that the numbers 1 to 35 are listed, and you pick 5.

a. How many unique tickets are possible? b. How many will win first place? c. How many possible second place tickets exist? d. How many tickets are possible which contain exactly 2 correct numbers? e. How many tickets are possible which contain no correct numbers? f. How many tickets are possible which contain at least 1 correct number?

26. In Powerball, one ticket is $2, and Powerball is 5 white balls from 1 to 69, with the 1 red “Power” ball

chosen from 1 to 26. a. How much money would you need in order to buy every single Powerball ticket? (a strategy that

would guarantee you win the grand prize) b. In Powerball, how many different tickets could there be that match all 5 white balls, but miss the

Powerball? (This wins $1,000,000) c. You can win $100 in Powerball by matching 3 white numbers and the Powerball. How many

different tickets meet this requirement? 27. Consider a box of 50 batteries, with 3 defective. Take a sample of 10 batteries. (While this seems

strange, it is like a lottery!) a. How many ways can the sample be chosen? b. How many samples have exactly 1 defective battery? c. How many have no defective batteries?


Wrap-up and look back:

28. What is the hardest part in the counting section for you? Why? 29. http://www.consumerfed.org/pdfs/Financial_Planners_Study011006.pdf A study released over 10

years ago showed that over 20% of Americans said that the most practical way for them to accumulate several hundred thousand dollars is to “win the lottery.” Based on this section, how likely do you believe it is that you could even win $100 in Powerball? Explain your reasoning.


1.14: Summary (Review)

1. What is a fallacy? Give 2 examples of different fallacies. 2. Simone said “When I eat bread, I have indigestion and when I eat normal pasta, I have indigestion. I

must be gluten intolerant.” Is Simone using inductive or deductive reasoning? Is her conclusion guaranteed?

3. Determine whether the sentence is a statement or not, then if it is a statement, find the truth value. Sentence Proposition Truth Value

a. Sunita Cooke is the Vice President of MiraCosta

College. Yes No True False

b. 17 is odd. Yes No True False

c. John Adams was the second president of the USA. Yes No True False

d. All grass needs water to grow. Yes No True False

e. 79 < 45 Yes No True False

f. x + 1 = 78 Yes No True False

g. Some computers cost more than $1,000. Yes No True False

4. Write the negation of the following statements:

a. All cartoons are done in black and white. b. Some cars cost $35,000. c. No drugs are bad for you.

d. 79 < 45 e. I am happy. f. Today is a good day.

5. Determine the truth values of the statements listed using a truth table.

p q qp p~ q~ qp qp ~ qp ~~ qp ~ qp ~~

a. T F

b. F F

c. T T

d. F T


6. When is a conditional statement like mp false? 7. What is the contrapositive of the following (put answer in symbols):

a. Lr ~ b. LP

8. Determine the truth values of the statements listed using a truth table. p q p~ q~ qp pq qp ~~ qp ~ pq ~~

a. T F

b. F F

c. T T

d. F T

9. Find an equivalent expression to the given statement (write your answer in English, not symbols).

a. If it is sunny, then we will go to the game.

b. If my sister has a boy, then we will not buy a new car.

10. What would it take to prove the following statements false? a. All rich people own a home.

b. Some turtles eat lettuce.

c. No English class is easy.

11. Use your negation knowledge to negate the following statements (answer as an English sentence): a. If I am hungry, then I drive well. b. All teens use cell phones. c. Some sheep are red.

d. It is cloudy or 5 + 5 = 10. e. I am sad and you do not wear hats. f. No people like to read.

12. Use your negation knowledge to negate the following statements (answer with symbols): a. qp ~

b. wp ~~

c. qr ~~

d. qp ~~ 13. What does it mean for an argument to be valid? 14. Is this argument valid? Explain your reasoning.

1. All cats climb trees. 2. Jonathan doesn’t climb trees. c. Jonathan is a cat.

15. What is the conclusion to this valid argument?

a. All cats like to eat mice. b. Jenny doesn’t like to eat mice. c.


16. Determine the truth values of these statements if you know that p is false, q is false, and r is true. If you see any other letter for a statement, it could be true or false… we don’t know. Your answer could be true, false, or need more information. Explain your answer.

Statement Truth Value

a. pw ~ True False Need More Information

b. wp True False Need More Information

c. qw True False Need More Information

d. wqp True False Need More Information

e. prq True False Need More Information

f. rqw ~ True False Need More Information

g. wr ~ True False Need More Information

17. Kylo Ren knows this theorem: “If a number ends with a 6, then it is divisible by 2.”

a. What (if anything) can you conclude from the theorem about the number 7,890? b. What (if anything) can you conclude from the theorem about the number 7,896? c. What (if anything) can you conclude from the theorem about the number 71,911,345? d. Kylo tells you that he sees a number that is not divisible by 2. Based on the theorem, what could he

conclude? 18. A Venn diagram is labeled. Determine what quantified

statement you could make from the diagram. Then write letters in the appropriate places in your diagram.

I. “X” represents a ballerina. II. “Z” represents a math teacher in bad shape.

III. “M” represents a runner (who doesn’t dance) but is in good shape.

19. Shade the appropriate regions for each type of graph. a. Shade BA . b. Shade B . c. Shade BA .

Dancers

In good shape

U A B

U B A

U A B


20. Based on the screenshot from Minesweeper,

determine which boxes are on “bombs” and which boxes are on “numbers” and which boxes you don’t know yet.

21. Determine the solution for these MasterMind puzzles (full 4 locations for these, but only 4 colors).

Guess Location 1 Location 2 Location 3 Location 4 Clues 1 G Y B B WH-WH-WH

2 B R R R WH

3 Y Y B Y BL-BL-WH

CODE R G Y B

R G Y B

R G Y B

R G Y B

22. Fill in a 2-circle Venn diagram with the numbers stated,

and answer any questions. 79 people came to the bike shop. 29 needed tires, 38 needed a tune-up, and 20 didn’t need either of these services. How many people wanted both tires and a tune-up?

23. 78 people buy 1 raffle ticket each. 4 winning tickets will be drawn. How many different ways can the

prizes be awarded if a. the first prize is $100, the second is $50, the third is $20, and the fourth is $10? b. if each prize is $40?

24. Determine the number of ways to arrange the objects listed:

a. How many ways can you rank the top 3 albums, in order, from a set of 12 albums? b. There are 9 different colored pegs and you are making a 4 color code where you can repeat options.

How many “codes” can be made? c. There are 9 different colored pegs and you are making a 4 color code where you cannot repeat

options. How many “codes” can be made? d. How many ways are there to rearrange the letters in SASSY? e. Out of a group of 10 men and 15 women, a committee of 5 is created. How many ways can the

committee be formed with: i. Any arrangement of men and women?

ii. Exactly 2 men? iii. Exactly 4 women?

iv. No men? v. At least 1 man?

vi. All men?

U

math fundamentals for statistics (math 95)home.miracosta.edu/bpickett/16sp m 95 and 126 and 131/math...

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