1. Chapter 12 Sec 1 The Counting Principle

2. 2 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Independent Events An outcome is the result of a single trial. An outcome is the result of a single trial. Flipping a coin. Flipping a coin. The set of all possible outcomes is the sample set. The set of all possible outcomes is the sample set. An event consists of one or more outcomes. An event consists of one or more outcomes. The choices of letters on a license plate are called independent events because each letter does not affect the choices for the others. The choices of letters on a license plate are called independent events because each letter does not affect the choices for the others.

3. 3 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 1 A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayo. How many different combinations of bread and spread are possible. A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayo. How many different combinations of bread and spread are possible. Method 1: Tree BreadWWhR SpreadB Mu Ma B Mu Ma B Mu Ma PossibleWB WMu WMa WhB WhMu WhMa RB RMu RMa There are 9 possible outcomes Method 2: Table Spreads ButterMustardMayo Breads WhiteWBWMuWMa WheatWhBWhMuWhMa RyeRBRMuRMa

4. 4 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 The Fundamental Counting Principle

5. 5 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 2 Many answering machines allow owners to call home and get their messages by entering a 3-digit code. How many codes are possible? 10 x 10 x 10 = 100 If the code is just 2 digits? 10 x 10 = 100

6. 6 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Dependent Events With dependent events the outcome of one event does affect the outcome of another. With dependent events the outcome of one event does affect the outcome of another. The fundamental Counting principle applies to dependent events as well. The fundamental Counting principle applies to dependent events as well.

7. 7 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 1 Carlin wants to take 6 different classes next year. Assume that each class is offered each period, how many different schedules could he have? Carlin wants to take 6 different classes next year. Assume that each class is offered each period, how many different schedules could he have? When he schedules a class that class won’t be available to for the following periods. So, When he schedules a class that class won’t be available to for the following periods. So, Per 1 – 6 ChoicesPer 2 – 5Per 3 – 4 …Per 6 – 1 Per 1 – 6 ChoicesPer 2 – 5Per 3 – 4 …Per 6 – 1 6 x 5 x 4 x 3 x 2 x 1 or 6! = 720 Choices

8. 8 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Sum it up (get it, counting principles…sum…never mind)

9. Chapter 12 Sec 2 Permutations and Combinations

10. 10 of 25 Algebra 2 Chapter 12 Sections 1 thru 3Permutations When a group of objects or people are arranged in a certain order, the arrangement is called a permutation. In permutation, the order of the objects is very important When a group of objects or people are arranged in a certain order, the arrangement is called a permutation. In permutation, the order of the objects is very important The arrangement in a line is call linear permutation. The arrangement in a line is call linear permutation.

11. 11 of 25 Algebra 2 Chapter 12 Sections 1 thru 3Permutation 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!.

12. 12 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 1 Eight people enter the Best Pie contest. How many ways can blue, red and green ribbons be awarded. n = 8 and r = 3 P(n, r) =

13. 13 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Permutation with Repetitions How many different ways can the letters of the word BANANA be arranged? How many different ways can the letters of the word BANANA be arranged? There are 2 Ns and 3 As. There are 2 Ns and 3 As.

14. 14 of 25 Algebra 2 Chapter 12 Sections 1 thru 3Combinations An arrangement of objects in which order is not important is called a combination. An arrangement of objects in which order is not important is called a combination. The number of combinations of n object take r at a time is written C(n, r) or n C r.. The number of combinations of n object take r at a time is written C(n, r) or n C r..

15. 15 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 2 Five cousins at a family reunion decide that three of them will go pick up a pizza. How many ways can they choose the three to go?

16. 16 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 3 Six cards are drawn from a standard deck of cards. How many hands consist of two hearts and four spades? Hearts - C(13,2) Spades - C(13,4)

17. Chapter 12 Sec 3 Probability

18. 18 of 25 Algebra 2 Chapter 12 Sections 1 thru 3Probability The probability of an event is a ratio that measures the chances of the event occurring. The probability of an event is a ratio that measures the chances of the event occurring. A desired outcome is called a success. A desired outcome is called a success. Any other outcome is called a failure Any other outcome is called a failure

19. 19 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 The Proverbial Coin Toss When three coins are tossed, what is the probability that all three will be heads? First CoinHT 2 nd CoinH T H T And 3 rd 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

20. 20 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 2 Roman has a collection of 26 books – 16 are fiction and 10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability that he will chooses 4 fiction and 4 nonfiction? First determine how many ways you can get 4 of each. To find s, use the Fundamental Counting Principle 1820 x 210 = 382,200 Total of s + f SO…

21. 21 of 25 Algebra 2 Chapter 12 Sections 1 thru 3Odds Another way t measure chance is with odds. Another way t measure chance is with odds. The odds that an event will occur can be expressed as a ratio of the number of successes to the number of failures. The odds that an event will occur can be expressed as a ratio of the number of successes to the number of failures.

22. 22 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 3 According to the US National Center 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 females, the chances are about 17 in 20. a. What are the odds of a male living to be at least 65? 3 out of 4 males will make it so successes = 3 4 – 3 will equal failures = 1 odds are 3:1 b. What are the odds of a female living to be at least 65? 17 out of 20 females will make it so successes = 17 20 – 17 will equal failures = 3 odds are 17:3

23. 23 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Probability Distribution Many experiments have numerical results. Many experiments have numerical results. A random variable is the numerical outcome of a random event. A random variable is the numerical outcome of a random event. A probability distribution for a random variable is a function that maps the sample space to the probabilities of the outcomes. A probability distribution for a random variable is a function that maps the sample space to the probabilities of the outcomes. ie The table below illustrates the probability distribution for casting a die. ie The table below illustrates the probability distribution for casting a die.

24. 24 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Example 4 Suppose two dice are rolled. The table and the relative-frequency histogram show the distribution of the sum of the numbers rolled. a. Use the graph to determine which outcome is most likely What is the probability? b. Use the table to find P(S = 9). What other 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. c. What are the odds of rolling a sum of 7? c. Identify s and f. Odds: s:f or 1:5

25. 25 of 25 Algebra 2 Chapter 12 Sections 1 thru 3 Daily Assignment Chapter 12 Sections 1 – 3 Study Guide (SG) Pg 157 – 162 All