Chapter 12 Sec 1

Chapter 12 Sec 1Chapter 12 Sec 1

The Counting The Counting PrinciplePrinciple

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Algebra 2 Chapter 12 Sections 1 thru 3

Independent EventsIndependent Events

• An An outcome outcome is the result of a single trial.is the result of a single trial.

• Flipping a coinFlipping a coin. .

• The set of all possible outcomes is the The set of all possible outcomes is the sample set.sample set.

• AnAn event event consists of one or more outcomes.consists of one or more outcomes.

• The choices of letters on a license plate are The choices of letters on a license plate are called called independent events independent events because each letter because each letter does not affect the choices for the others.does not affect the choices for the others.

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Example 1Example 1• A sandwich menu offers customers A sandwich menu offers customers a choice of white, wheat, or rye bread a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayo. How many with one spread chosen from butter, mustard, or mayo. How many different combinations of bread and spread are possible.different combinations of bread and spread are possible.

Method 1: Tree

Bread W Wh R

Spread B Mu Ma B Mu Ma B Mu Ma

Possible WB WMu WMa WhB WhMu WhMa RB RMu RMa

There are 9 possible outcomes Method 2: Table Spreads

Butter Mustard Mayo

BreadsWhite WB WMu WMa

Wheat WhB WhMu WhMa

Rye RB RMu RMa

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The Fundamental Counting Principle The Fundamental Counting Principle

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Example 2Example 2

Many answering machines allow owners to call Many answering machines allow owners to call home and get their messages by entering a 3-digit home and get their messages by entering a 3-digit code. How many codes are possible?code. How many codes are possible?

10 x 10 x 10 = 10010 x 10 x 10 = 100

If the code is just 2 digits?If the code is just 2 digits?

10 x 10 = 10010 x 10 = 100

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Dependent EventsDependent Events

• With With dependent events dependent events the outcome of one the outcome of one event event does does affect the outcome of another.affect the outcome of another.

• The fundamental Counting principle applies The fundamental Counting principle applies to dependent events as well.to dependent events as well.

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Example 1Example 1• Carlin wants to take 6 different Carlin wants to take 6 different classes next year. Assume that each class is offered classes next year. Assume that each class is offered each period, how many different schedules could he each period, how many different schedules could he have?have?

• When he schedules a class that class won’t be When he schedules a class that class won’t be available to for the following periods. So,available to for the following periods. So,

• Per 1 – 6 ChoicesPer 1 – 6 Choices Per 2 – 5Per 2 – 5 Per 3 – 4 …Per 3 – 4 … Per 6 – 1 Per 6 – 1

6 6 xx 5 5 xx 4 4 xx 3 3 xx 2 2 xx 1 or 6! = 1 or 6! =

720 Choices720 Choices

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Sum it up Sum it up (get it, counting principles…sum…never (get it, counting principles…sum…never mind)mind)


Permutations and Permutations and CombinationsCombinations

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PermutationsPermutations

• When a group of objects or people are arranged in When a group of objects or people are arranged in a certain order, the arrangement is called a a certain order, the arrangement is called a permutationpermutation. In permutation, the . In permutation, the orderorder of the of the objects is objects is veryvery important important

• The arrangement in a line is call The arrangement in a line is call linear linear permutation.permutation.

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PermutationPermutation• Notice that is the first 4 factors of 7!.Notice that is the first 4 factors of 7!.

You can rewrite this in terms of 7!. You can rewrite this in terms of 7!.

4567 123

1234567

123

1234567

!3

!7

4567

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Example 1Example 1Eight people enter the Best Pie contest. How Eight people enter the Best Pie contest. How many ways can blue, red and green ribbons many ways can blue, red and green ribbons be awarded.be awarded.

n = n = 8 and 8 and r =r = 3 3

PP((n, rn, r) = ) = !5

!8

!38

!8

!5

!5678

336678

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Permutation with RepetitionsPermutation with Repetitions

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

• There are 2 Ns and 3 As.There are 2 Ns and 3 As.

!3!2

!6

!!

!

qp

n

!312

!3456

602

120

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CombinationsCombinations• An arrangement of objects in which order is An arrangement of objects in which order is notnot

important is called a important is called a combinationcombination. .

• The number of combinations of The number of combinations of nn object take object take r r at at a time is written a time is written CC((n, rn, r) or ) or nnCCrr....

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Example 2Example 2Five cousins at a family reunion decide that three Five cousins at a family reunion decide that three of them will go pick up a pizza. How many ways of them will go pick up a pizza. How many ways can they choose the three to go?can they choose the three to go?

!3!35

!53,5

C

!3!2

!5

!312

!345

!312

!345

102

20

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Example 3Example 3Six cards are drawn from a standard deck of cards. Six cards are drawn from a standard deck of cards.

How many hands consist of two hearts and four How many hands consist of two hearts and four spades?spades?

Hearts - CHearts - C(13,2) Spades - (13,2) Spades - CC(13,4)(13,4)

!4!9

!13

!2!11

!134,132,13 sh CC

770,5571578


ProbabilityProbability

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Probability Probability • The The probabilityprobability of an event is a ratio that of an event is a ratio that

measures the chances of the event occurring.measures the chances of the event occurring.

• A desired outcome is called a A desired outcome is called a successsuccess. .

• Any other outcome is called a Any other outcome is called a failurefailure

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The Proverbial Coin TossThe Proverbial Coin Toss

When three coins are tossed, what is the probability When three coins are tossed, what is the probability

that all three will be heads?that all three will be heads?

12.5%or 8

1

71

1)(

sP

First Coin H T

2nd Coin H T H T

And 3rd H T H T H T H T

Possible outcomes HHH HHT HTH HTT THH THT TTH TTT

8 Possible 1 Success and 7 Failure

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Example 2Example 2 Roman has a collection of 26 books – 16 Roman has a collection of 26 books – 16 are fiction and 10 are nonfiction. He randomly chooses 8 are fiction and 10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability books to take with him on vacation. What is the probability that he will chooses 4 fiction and 4 nonfiction?that he will chooses 4 fiction and 4 nonfiction?First determine how many ways you can get 4 of each.First determine how many ways you can get 4 of each.

To find To find s, s, use the Fundamental Counting Principle use the Fundamental Counting Principle 1820 1820 xx 210 = 382,200 210 = 382,200

Total of Total of s + f s + f

SO…SO…

210!4!6

!104,10 and 1820

!4!12

!164,16 CC

1,562,275 !8!18

!268,26 C

%24or ...24464.275,562,1

200,382

fs

s

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OddsOdds• Another way t measure chance is with Another way t measure chance is with odds. odds.

• The odds that an event will occur can be The odds that an event will occur can be expressed as a ratio of the number of successes expressed as a ratio of the number of successes to the number of failures.to the number of failures.

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Example 3Example 3According to the US National Center According to the US National Center for Health Stats, the chances of a male born in 1990 for Health Stats, the chances of a male born in 1990 living to be at least 65 years of age are about 3 in 4. For living to be at least 65 years of age are about 3 in 4. For females, the chances are about 17 in 20.females, the chances are about 17 in 20.

a. What are the odds of a male living to be at least 65?a. What are the odds of a male living to be at least 65?

3 out of 4 males will make it so successes = 33 out of 4 males will make it so successes = 3

4 – 3 will equal failures = 1 odds are 3:14 – 3 will equal failures = 1 odds are 3:1

b. What are the odds of a female living to be at least 65?b. What are the odds of a female living to be at least 65?

17 out of 20 females will make it so successes = 1717 out of 20 females will make it so successes = 17

20 – 17 will equal failures = 3 odds are 17:320 – 17 will equal failures = 3 odds are 17:3

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Probability DistributionProbability Distribution• Many experiments have numerical results.Many experiments have numerical results.• A A random variable random variable is the numerical outcome is the numerical outcome

of a random event.of a random event.• A A probability distribution probability distribution for a random for a random

variable is a function that maps the sample variable is a function that maps the sample space to the probabilities of the outcomes.space to the probabilities of the outcomes.• ie The table below illustrates the probability distribution ie The table below illustrates the probability distribution

for casting a die.for casting a die.

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Example 4Example 4 Suppose two dice are rolled. The Suppose two dice are rolled. The table and the relative-frequency histogram show the table and the relative-frequency histogram show the distribution of the sum of the numbers rolled.distribution of the sum of the numbers rolled.

a. Use the graph to determine which outcome a. Use the graph to determine which outcome is most likely What is the probability?is most likely What is the probability?b. Use the table to find P(S = 9). What other b. Use the table to find P(S = 9). What other sum has the same probability. sum has the same probability.

a. The greatest probability is and the most likely outcome is a sum of 7.

b. The P(9) is and the other outcome is 5.

6

1

9

1

c. What are the odds of rolling a sum of 7? c. What are the odds of rolling a sum of 7?

c. Identify s and f.

Odds: s:f or 1:5

5,1...6

1)7(

fs

fs

sP

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Daily AssignmentDaily Assignment

• Chapter 12 Sections 1 – 3 Chapter 12 Sections 1 – 3 • Study Guide (SG)Study Guide (SG)

• Pg 157 – 162 AllPg 157 – 162 All

Documents

Chapter 12 Sec 1