19.2 Permutations and Probability Combinations and Probability.

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2 19.2 Permutations and Probability Combinations and Probability. Use permutations and combinations to compute probabilities of compound events and solve problems. When are permutations useful in calculating probability? What is the difference between a permutacion and a combination?

3 What is a permutation? A permutation is a selection of objects from a group in which order is important. For example, there are 6 permutations of the letters A, B, and C. Fundamental Counting Principle: If you have A ways of doing event 1, B ways of doing event 2, and C ways of event 3, then you can find the total number of outcomes by multiplying: A B C This principle is difficult to explain in words. To find the total number of outcomes for the scenario, multiply the total outcomes for each individual event. For example: 3 choices of sandwiches 3 choices of sides 2 choices of drinks = 18 total outcomes

4 There are 7 members in a club. Each year the club elects a president, a vice president, and a treasurer = =

5 = = 210

6 What is factorial? Finding a Probability Using Permutations 4! Is usually pronounced 4 factorial, but some people even say 4 shriek or 4 bang That is, n! is the product of n and all the positive integers less than n. Note that 0! is defined to be 1.

7 Permutations Another way to write it GC have a built-in function that calculates permutations. For example, to find 10P4, first enter 10. Then press MATH and use the arrow keys to choose the PRB menu. Select 2:nPr and press ENTER. Now enter 4 and then press ENTER to see that P(10,4) = 5040.

8 Example 1 Use permutations to find the probabilities. 5 digits 10 digits ! 5! 30, ! 0! all even digits all even digits 30,

9 Explain 2 Finding the Number of Permutations with Repetition Up to this point, the problems have focused on finding the permutations of distinct objects. If some of the objects are repeated, this will reduce the number of permutations that are distinguishable. For example, here are the permutations of the letters A, B, and C. Permutations with Repetition The number of different permutations of n objects where one object repeats a times, a second object repeats b times, and so on is n! a! b! Example 2 Find the number of permutations. A. How many different permutations are there of the letters in the word ARKANSAS. Answer There are 8 letters in the word, and there 3 A s and 2 S s. So the number of permutations of the letters in the word ARKANSAS is 8! 3!2! = 3360

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11 Explain 3 Finding a Probability Using Permutations with Repetition (pg. 965) Example 3 The school jazz band has 4 boys and 4 girls, and they are randomly lined up for a yearbook photo. (Permutations with repetition can be used to find probablilities.) A Find the probability of getting an alternating boy-girl arrangement. The sample space S consists of permutations of 8 objects, with 4 boys and 4 girls. Event A consists of permutations that alternate boy-girl or girl-boy. The possible permutations are BGBGBGBG and GBGBGBGB. The probability of getting an alternating boy-girl arrangement is B Find the probability of getting all of the boys grouped together. The sample space S consists of permutations of, 8 students with. 4 boys and 4 girls 8! 4!4! 70 Event A consists of permutations with all 4 boys in a row. The possible permutations are BBBBGGGG, GBBBBGGG, GGBBBBGG, GGGBBBBG, and GGGGBBBB. n(a) = 5 n A The probability of getting all the boys grouped together is P A = n S = 5 70 = 1 14

12 What A combination is a combination? is a selection of objects from a group in which order is unimportant. A restaurant has 8 different appetizers on the menu, as shown in the table. They also offer an appetizer sampler, which contains any 3 of the appetizers served on a single plate. How many different appetizer samplers can be created? The order in which the appetizers are selected does not matter. Nachos Chicken Wings Answer Chicken Quesadilla Potato Skins Beef Chili Vegetarian Egg Rolls Soft Pretzels Guacamole Dip A. Find the number appetizer samplers that are possible if the order of selection does matter. This is the number of permutations of 8 objects taken 3 at a time. 8! 8! ! B. Find the number of different ways to select a particular group of appetizers. This is the number of permutations of 3 objects. 3! 3! ! C. To find the number of possible appetizer samplers if the order of selection does not matter, divide the answer to part A by the answer to part B. So the number of appetizer samplers that can be created is = 56

13 Finding a Probability Using Combinations The number of combinations of the 8 objects taken 3 at a time is This can be generalized as follows Here are some other notations

14 Example 1 Find each probability. B. There are 52 cards in a standard deck, 13 in each of 4 suits: clubs, diamonds, hearts, and spades. Five cards are randomly drawn from the deck. What is the probability that all five cards are diamonds? Answer ! 2, 598, 960 5!47! ! 5!8! ,598,960 66,640

15 Explain Finding a Probability Using Combinations and Addition Sometimes, counting problems involve the phrases at least or at most. For these problems, combinations must be added. B. Three number cubes are rolled and the result is recorded. What is the probability that at least 2 of the number cubes show 6? Answer The number of outcomes in the sample space S can be found by using the Fundamental Counting Principle since each roll can result in 1, 2, 3, 4, 5, or Let A be the event that at least 2 number cubes show 6. This is the sum of 2 events 2, number cubes showing 6 or. 3 number cubes showing 6 The event of getting 6 on 2 number cubes occurs since 5 times there are 5 possibilities for the other number cube. The probability of getting a 6 at least twice in 3 rolls is

16 Classwork and Homework Pg exercises 1-27 and pg Exercises 1-26 Ready for a QUIZ next class?