algebra-2 counting and probability. quiz 10-1, 10-2 1. which of these are an example of a...

Algebra-2Algebra-2

Counting and ProbabilityCounting and Probability

Quiz 10-1, 10-2Quiz 10-1, 10-21. Which of these are an example of a “descrete” set of data?1. Which of these are an example of a “descrete” set of data?

11 , 9 7, 5, x :such that . xa 11 x 5 :such that . xb

2.2. Make a “tree diagram” showing all the ways the letters Make a “tree diagram” showing all the ways the letters ‘ ‘x’, ‘y’, and ‘z’ can be arranged in order.x’, ‘y’, and ‘z’ can be arranged in order.

3. You are paying for groceries at the store. You have the 3. You are paying for groceries at the store. You have the following bills: $100, $50, $20, $10, $5, $2, and $1.following bills: $100, $50, $20, $10, $5, $2, and $1.

)!(!!rnr

nCrn

What are number of different What are number of different sumssums of money that you can pull of money that you can pull out of your wallet if you pull out 3 bills without looking?out of your wallet if you pull out 3 bills without looking?

)!(!rnnPrn

Counting Counting How many ways can you arrange the letters How many ways can you arrange the letters

‘x’, ‘y’, and ‘z’ in order?‘x’, ‘y’, and ‘z’ in order? ___ ___ ______ ___ ___33 22 11

Why isn’t it ? Why isn’t it ? ___ ___ ______ ___ ___33 33 33

You can’t re-use a letter in this situation.You can’t re-use a letter in this situation.

Can you think of a situation where you could Can you think of a situation where you could re-usere-use a number or a letter? a number or a letter?

Phone numbersPhone numbers Social security numbersSocial security numbersLicense platesLicense plates

Go to demoGo to demo

VocabularyVocabulary

““arranging without replacementarranging without replacement: when you use an item in: when you use an item in the arrangement, it is “used up” and can’t be used again.the arrangement, it is “used up” and can’t be used again.

““arranging with replacementarranging with replacement: when an item is used in one: when an item is used in one position in an arrangement, it can be used again in anotherposition in an arrangement, it can be used again in another position in the arrangement.position in the arrangement.

Think of arranging people in a line. Once a person is in theThink of arranging people in a line. Once a person is in the front of the line, he cannot also be in the back of the linefront of the line, he cannot also be in the back of the line at the same time. at the same time.

Think of arranging numbers and Letters on a license plate: Think of arranging numbers and Letters on a license plate: the previous number or letter can be used again.the previous number or letter can be used again.

Effect on Muliplication Principle of Effect on Muliplication Principle of countingcounting ( (Product of the # of options for each step)Product of the # of options for each step)

arranging without replacementarranging without replacement::

arranging with replacementarranging with replacement::

Arranging 3 numbers on a licence plate. Arranging 3 numbers on a licence plate.

Arranging 3 people in a line. Arranging 3 people in a line. 1*2*3

10*10*10

FactorialFactorial

Your turn:Your turn:Which is it (with or without replacement) for:Which is it (with or without replacement) for:

1. 1. Assigning 3 committee members to the positions of:Assigning 3 committee members to the positions of: “ “Pres”, “Vice-Pres”, and “Secretary”Pres”, “Vice-Pres”, and “Secretary”

2. 2. The total number of social security numbers with 9 The total number of social security numbers with 9 digits.digits.

Using the Multiplication Using the Multiplication PrinciplePrincipleIf a license plate has three letters followed by three If a license plate has three letters followed by three

numerical digits. Find the number of different numerical digits. Find the number of different license plates that could be formed if there is no license plates that could be formed if there is no restriction on the letters or digits that can be used.restriction on the letters or digits that can be used.

L L L # # # L L L # # #

How many possibilities for the 1How many possibilities for the 1stst position (letter)? position (letter)?26 26

Using the Multiplication Using the Multiplication PrinciplePrinciple

L L L # # # L L L # # #

How many possibilities for the 2How many possibilities for the 2ndnd position? position?

26 26 * 26 * 26


L L L # # # L L L # # #

How many possibilities for the 3How many possibilities for the 3rdrd position? position?

26 26 * 26 * 26 * 26 * 26


L L L # # # L L L # # #

How many possibilities for the 4How many possibilities for the 4thth position (number)? position (number)?

26 26 * 26 * 26 * 26 * 26 * 10 * 10


L L L # # # L L L # # #

How many possibilities for the 5How many possibilities for the 5thth position? position?

26 26 * 26 * 26 * 26 * 26 * 10 * 10 * 10 * 10


L L L # # # L L L # # #

How many possibilities for the 6How many possibilities for the 6thth position? position?

26 26 * 26 * 26 * 26 * 26 * 10 * 10 * 10 * 10 * 10 * 10

Total number of distinct license plates = 676,000 Total number of distinct license plates = 676,000

Your Turn:Your Turn:3. 3. How many distinct license plates can be made using How many distinct license plates can be made using 6 digits (numerals 0 – 9)? (# # # # # #)6 digits (numerals 0 – 9)? (# # # # # #)

4. 4. How many distinct license plates can be made using How many distinct license plates can be made using 2 digits (numerals 0 – 9) and 4 letters ( A – Z) ?2 digits (numerals 0 – 9) and 4 letters ( A – Z) ? (# # L L L L)(# # L L L L)

1,000,000 1,000,000

45,697,600 45,697,600

Your Turn:Your Turn:Count the number of different 8-letter “words” (groups of 8 letters) that Count the number of different 8-letter “words” (groups of 8 letters) that can be formed using the letters in the word COMPUTER. can be formed using the letters in the word COMPUTER.

Each permutation of the 8 letters forms a different Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.word. There are 8! = 40,320 such permutations.

5. 5.

What if two of the letters are What if two of the letters are the same?the same?

Count the number of different 4-letter “words” that can be formed using the letters in the word Count the number of different 4-letter “words” that can be formed using the letters in the word “ “WWAAAAG”. G”.

Let “Let “AA” be the 1” be the 1stst A. A.

Let “Let “AA” be the 2” be the 2ndnd A. A.

What’s the difference between What’s the difference between AAAAWGWG and and AAAAWGWG??

There’s no difference!! They are not There’s no difference!! They are not distinguishabledistinguishable from each other. So we really have “from each other. So we really have “double counteddouble counted” ” a bunch of words. a bunch of words.

WAAGWAAG”. ”.

What if two of the letters are What if two of the letters are the same?the same?Count the number of different 4-letter “words” that can be formed using the letters in the word Count the number of different 4-letter “words” that can be formed using the letters in the word

“ “WAAG”. WAAG”.

AAAAWGWG ( (AAAAWGWG) is one example of double counting.) is one example of double counting.

To remove the “To remove the “double countingdouble counting” we must” we must divide out the number of possible ways to divide out the number of possible ways to permutate permutate A A andand AA

AAWWAAGG ( (AAWWAAGG) is another example of double counting.) is another example of double counting.

We must divide by 2!.We must divide by 2!.

What if three of the letters are What if three of the letters are the same?the same?

Count the number of different 5-letter “words” that can be formed using the letters in the word Count the number of different 5-letter “words” that can be formed using the letters in the word “ “WAAAG”. WAAAG”.

AAAAAAWGWG AAAAAAWGWG AAAAAAWGWG AAAAAAWGWG AAAAAAWGWG AAAAAAWGWG

To remove the “To remove the “double countingdouble counting” we must” we must divide out the number of ways to permutate divide out the number of ways to permutate A, A, AA and and AA

These are all examples of the same word and have beenThese are all examples of the same word and have been “ “double counted”.double counted”.

We must divide by 3!We must divide by 3!

AAAWGAAAWG AAAWGAAAWG AAAWGAAAWG AAAWGAAAWG AAAWGAAAWG AAAWGAAAWG

“WWAAAAAAG”G”.

Distinguishable Distinguishable PermutationsPermutations

We must “divide out” the permutations of the same object We must “divide out” the permutations of the same object that result in that result in indistinguishableindistinguishable arrangements. arrangements.

!5!*4!*3!12

If a set 12 items to be permutated has 3 objects of one kind,If a set 12 items to be permutated has 3 objects of one kind, and 4 objects of another kind, and 5 objects of another kind,and 4 objects of another kind, and 5 objects of another kind, then the number of then the number of distinguishabledistinguishable ways to arrange the 12 ways to arrange the 12 items is:items is: 720,27

Distinguishable Distinguishable PermutationsPermutationsIn general, we find the number of distinguishable permutationsIn general, we find the number of distinguishable permutations when using some elements that are indistinguishablewhen using some elements that are indistinguishable as follows: as follows:

!!*!*!CBA

N

If a set If a set NN items to be permutated has items to be permutated has AA objects of one kind, objects of one kind, and and BB objects of another kind, and objects of another kind, and CC objects of another kind, objects of another kind, and and AA + + BB + + CC = = NN then the number of then the number of distinguishabledistinguishable ways ways to arrange the to arrange the N N items is: items is:

Your Turn:Your Turn:You have the following bills in your wallet: You have the following bills in your wallet:

three $20’s, four $10’s, five $5’s, and six $1’s three $20’s, four $10’s, five $5’s, and six $1’s

What is the number of distinct ways you could pay What is the number of distinct ways you could pay out the bills one at a time?out the bills one at a time?

6. 6.

!6!*5!*4!*3!18

080,594,514

CountingCountingHow many 5 card hands are there with all face cards (king, How many 5 card hands are there with all face cards (king,

queen, jack).queen, jack).

Which is it?Which is it?

This tells you the hands all have 5 face cards. So howThis tells you the hands all have 5 face cards. So how many arrangements are there when taking 12 cards and many arrangements are there when taking 12 cards and picking 5 ?picking 5 ?

PermutationPermutation: (different order : (different order counted separately) counted separately)CombinationCombination: (different order : (different order notnot counted separately) counted separately)

792512 C

Your Turn:Your Turn:How many 5 card hands are there with How many 5 card hands are there with nono face cards? face cards?7. 7.

540C 008,658

CountingCountingSometimes there are more than one condition that must be met.Sometimes there are more than one condition that must be met.

How many 5 card hands have all 5 cards the same suite How many 5 card hands have all 5 cards the same suite (hearts, diamonds, spades, clubs).(hearts, diamonds, spades, clubs).

CombinationCombination: (different order : (different order notnot counted separately) counted separately)

14C11stst we must pick the suite: we must pick the suite:

22ndnd we must pick the 5 cards from that suite: we must pick the 5 cards from that suite: 513CBy the multipication principle: total number hands is:By the multipication principle: total number hands is:

14C 513* C 5148

CountingCountingHow many 5 card hands have How many 5 card hands have exactlyexactly 2 aces ? 2 aces ?

CombinationCombination: (different order : (different order notnot counted separately) counted separately)

24C11stst we must pick the 2 aces: we must pick the 2 aces:

22ndnd we must pick the other 3 cards we must pick the other 3 cards: (if the hand has : (if the hand has exactly 2 aces, then we must not include the other two exactly 2 aces, then we must not include the other two aces as possible picks)aces as possible picks)

348C

By the multipication principle: total number of hands is:By the multipication principle: total number of hands is:

24C 348* C 103776

Your Turn:Your Turn:How many 5 card hands are there with How many 5 card hands are there with two fivestwo fives and and two sixestwo sixes??8. 8.

24C 158424* C 144* C

Hint: (1) pick the 2 fives, (2) pick the 2 sixes, (3) Hint: (1) pick the 2 fives, (2) pick the 2 sixes, (3) Pick the last card. Use the multiplication rule.Pick the last card. Use the multiplication rule.

ProbabilityProbability““What’s the chance of something What’s the chance of something happening?”happening?”

““There is a 100% chance it will rain today.”There is a 100% chance it will rain today.”

““There is less than a 5% chance you will be picked.There is less than a 5% chance you will be picked.

Your turn:Your turn:Can probability be equal to 50%?Can probability be equal to 50%?

What is the largest number that a probability What is the largest number that a probability can be?can be?

What is the smallest number that a probability What is the smallest number that a probability can be?can be?

Can there be a (– 20)% chance something will Can there be a (– 20)% chance something will happen?happen?

9. 9.

10. 10.

11. 11.

12. 12.

ProbabilityProbabilityWhen When discussingdiscussing probability, you can use either probability, you can use either “%”, fraction, or the decimal equivalent. “%”, fraction, or the decimal equivalent.

“ “There is a 40% chance of thunderstorms today.”There is a 40% chance of thunderstorms today.”

In mathematicsIn mathematics, we convert % to the decimal , we convert % to the decimal equivalent or leave it in fraction form.equivalent or leave it in fraction form.

“ “The probability of rain today is 0.4.”The probability of rain today is 0.4.”

Theoretical ProbabilityTheoretical Probability

outcomes possible of # totalevent that achieve to waysof #)( eventP

bag in the marbles of # totalbag in the marbles red of #)( redP

The probability of an event occurring: The probability of an event occurring:

There are 4 different colored marbles in a bag (There are 4 different colored marbles in a bag (redred, , blueblue, , greengreen and and yellowyellow). What is the probability of ). What is the probability of pulling out a red one on the first try?pulling out a red one on the first try?

25.041)( redP

ExamplesExamplesThe probability of rolling a ‘5’ using one die.The probability of rolling a ‘5’ using one die.

166667.061)5( P

The probability of drawing a “king” from a deck of cards.The probability of drawing a “king” from a deck of cards.

0769.0131

524)( kingP



bag theofout letters 3 thedraw to waysof # totalC then B,A, draw to waysof #),,( CBAP

The challenge you have is The challenge you have is countingcounting the ways to the ways to achieve the event (sometimes called successful achieve the event (sometimes called successful

events) and then counting the total possible outcomes.events) and then counting the total possible outcomes.

3!1

What is the probility of pulling an What is the probility of pulling an AA, followed , followed by a by a BB, and then a, and then a C C out of a bag with the out of a bag with the

letters ‘letters ‘AA’, ‘’, ‘BB’, and ‘’, and ‘CC’ in it ?’ in it ?

3 permutate 31),,( CBAP 61.0

61

Your Turn:Your Turn:13. 13. What is the probability of picking the correct numberWhat is the probability of picking the correct number when someone asks you to pick a number from 1 to 10. when someone asks you to pick a number from 1 to 10.

14. 14. There are 2 red marbles and 3 green ones in a bag.There are 2 red marbles and 3 green ones in a bag. What is the probability of picking out a red marble on What is the probability of picking out a red marble on the first try? the first try?


Probability only works if the events are completely random.Probability only works if the events are completely random. Picking a committee using numbers out of a hat or a similarPicking a committee using numbers out of a hat or a similar random method of picking them is the only way thatrandom method of picking them is the only way that probability will work. probability will work.

Your Turn:Your Turn:15. 15. 10 people are trying to be selected for a 3 person 10 people are trying to be selected for a 3 person committee. You don’t know any of the people. What is committee. You don’t know any of the people. What is

the probality of you guessing whothe probality of you guessing who will be on the committee? will be on the committee?


16. 16. What is the probability of having a 5 card hand with a What is the probability of having a 5 card hand with a single pair of kings in it? single pair of kings in it?

Geometric Probability: ratio of Geometric Probability: ratio of areasareasAssumming that at an arrow randomly hits anywhere Assumming that at an arrow randomly hits anywhere in the four square area, what is the probability of in the four square area, what is the probability of hitting in the #1 square?hitting in the #1 square?

1 2

3 4

Since all squares have Since all squares have the the same areasame area, , and #1 is ¼ of the total and #1 is ¼ of the total area area probability is ¼. probability is ¼.

Geometric ProbabilityGeometric Probability: the area of each ring : the area of each ring is givenis given..

25

75

125

If an arrow will randomly hit anywhere inside of the red If an arrow will randomly hit anywhere inside of the red circle, what is the probability of hitting the center blue circle?circle, what is the probability of hitting the center blue circle?

area totalarea bluelight )( centerP

125752525)(

centerP

22525)( centerP

22525)( centerP

1.0)( centerP

Geometric ProbabilityGeometric Probability

25

75

125

17. 17. What is the probability of hitting the pink ring? What is the probability of hitting the pink ring?

18. 18. What is the probability of hitting either the pink or dark blue ring? What is the probability of hitting either the pink or dark blue ring?

End hereEnd here

Probability using combinations and Probability using combinations and permutations.permutations.

At the Roy High School Talent show 7 musicians are scheduled to perform. At the Roy High School Talent show 7 musicians are scheduled to perform. What is the probability that they will perform in alphabetical order of What is the probability that they will perform in alphabetical order of their last names (nobody has the same last name) ?their last names (nobody has the same last name) ?

There is only one order of performers that is in alphabetical order. There is only one order of performers that is in alphabetical order.


How many ways can you arrange 7 persons names in order?How many ways can you arrange 7 persons names in order?

Is this a permutation or combination?Is this a permutation or combination?

7 permutate 71)order alalphabetic( P

7!1)order alalphabetic( P

50401

0002.0

Probability using combinations and Probability using combinations and permutations.permutations.

At the Roy High School Talent show 7 musicians are scheduled to perform. At the Roy High School Talent show 7 musicians are scheduled to perform. 3 performers are girls and 4 are boys. What is the probability that all 3 3 performers are girls and 4 are boys. What is the probability that all 3 girls will be first? girls will be first?


How many ways can you get the first 3 performers to be girls?How many ways can you get the first 3 performers to be girls?

orderin people 7) (of 3 arrange to waysof #orderin girls 3 arrange toways#)1 girls 3( st P

3 choose 73 choose 3)1 girls 3( st P

37

1C

029.0

How many ways can you arrange 3 of 7 people in order?How many ways can you arrange 3 of 7 people in order?

Is this a permutation or combination?Is this a permutation or combination?

351

Your Turn:Your Turn:

19. 19. At the Roy High School Talent show 7 musicians are scheduled to At the Roy High School Talent show 7 musicians are scheduled to perform. They are: Bill, Brad, Bob, and Brody (boys) and Kylee, perform. They are: Bill, Brad, Bob, and Brody (boys) and Kylee, Kaylee, and Kyla (3 girls). What is the probability that 2 boys will be Kaylee, and Kyla (3 girls). What is the probability that 2 boys will be first? first?


orderin people 7) (of 2 arrange to waysof #orderin boys 4 of 2 arrange toways#)1 boys 2( st P

2 choose 72 choose 4)1 boys 2( st P

27

24

CC

29.0216

Your Turn:Your Turn:

20. 20. What is the probability of getting 4 aces in a randomly dealt hand of 4 What is the probability of getting 4 aces in a randomly dealt hand of 4 cards?cards?


52. ofout cards 4get to waysof #cards 4out aces 4get toways#)aces 4( P

4 choose 524 choose 4)acesr ( P

452

44

CC

0000037.0725,270

1

Your Turn:Your Turn:21. 21. A lottery uses numbers 1 thru 46. 6 numbers are drawn A lottery uses numbers 1 thru 46. 6 numbers are drawn

randomly. The order in which you choose the numbers randomly. The order in which you choose the numbers doesn’t matter. What is the probability of winning the lottery doesn’t matter. What is the probability of winning the lottery if you buy one ticket (assume nobody else picks the winning if you buy one ticket (assume nobody else picks the winning number) ? number) ?


numbers 46 of 6pick to waysof #numberscorrect 6 get the toways#)'# 6( sP

6 choose 466 choose 6)w( inP

646

1C

00000011.0819,366,9

1

How many ways can you get the 6 out of 6 correct numbers? How many ways can you get the 6 out of 6 correct numbers?

How many ways can you pick 6 of 46 numbers?How many ways can you pick 6 of 46 numbers?

Is picking 6 of 46 a permutation or a combination?Is picking 6 of 46 a permutation or a combination?

Cards:Cards:

What is the probability of getting 4 aces a randomly dealt hand of 5 cards?What is the probability of getting 4 aces a randomly dealt hand of 5 cards?


hands card 5 ofnumber totalcards 5 ofout aces 4 have to waysofnumber )aces 4( P

000015.0960,598,238

552

13844 C*C)aces 4(C

P

Your turn:Your turn:

17. 17. What is the probability of getting 3 aces and 2 kings from randomly What is the probability of getting 3 aces and 2 kings from randomly dealt hand of 5 cards?dealt hand of 5 cards?


hands card 5 ofnumber totalsk' 2 and aces 3 have to waysofnumber )sk' 2 aces, 3( P

000015.0960,598,224

552

2434 C*Cs)k' 2 ,aces 3(C

P

algebra-2 counting and probability. quiz 10-1, 10-2 1. which of these are an example of a...

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