Chapter Probability © 2010 Pearson Prentice Hall. All rights reserved 3 5

Section 5.5 Counting Techniques 5-2 © 2010 Pearson Prentice Hall. All rights reserved

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5-5 © 2010 Pearson Prentice Hall. All rights reserved For each choice of appetizer, we have 4 choices of entrée, and that, for each of these 2 4 = 8 choices, there are 2 choices for dessert. A total of = 16 different meals can be ordered. EXAMPLE Counting the Number of Possible Meals

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5-8 © 2010 Pearson Prentice Hall. All rights reserved A permutation is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol n P r represents the number of permutations of r objects selected from n objects.

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5-10 © 2010 Pearson Prentice Hall. All rights reserved In how many ways can horses in a 10-horse race finish first, second, and third? EXAMPLE Betting on the Trifecta The 10 horses are distinct. Once a horse crosses the finish line, that horse will not cross the finish line again, and, in a race, order is important. We have a permutation of 10 objects taken 3 at a time. The top three horses can finish a 10-horse race in

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5-12 © 2010 Pearson Prentice Hall. All rights reserved A combination is a collection, without regard to order, of n distinct objects without repetition. The symbol n C r represents the number of combinations of n distinct objects taken r at a time.

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5-14 © 2010 Pearson Prentice Hall. All rights reserved How many different simple random samples of size 4 can be obtained from a population whose size is 20? EXAMPLE Simple Random Samples The 20 individuals in the population are distinct. In addition, the order in which individuals are selected is unimportant. Thus, the number of simple random samples of size 4 from a population of size 20 is a combination of 20 objects taken 4 at a time. Use Formula (2) with n = 20 and r = 4: There are 4,845 different simple random samples of size 4 from a population whose size is 20.

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5-17 © 2010 Pearson Prentice Hall. All rights reserved How many different vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red? EXAMPLE Arranging Flags We seek the number of permutations of 10 objects, of which 5 are of one kind (white), 3 are of a second kind (blue), and 2 are of a third kind (red). Using Formula (3), we find that there are different vertical arrangements

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5-19 © 2010 Pearson Prentice Hall. All rights reserved In the Illinois Lottery, an urn contains balls numbered 1 to 52. From this urn, six balls are randomly chosen without replacement. For a $1 bet, a player chooses two sets of six numbers. To win, all six numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the probability of winning the lottery? EXAMPLE Winning the Lottery The probability of winning is given by the number of ways a ticket could win divided by the size of the sample space. Each ticket has two sets of six numbers, so there are two chances of winning for each ticket. The sample space S is the number of ways that 6 objects can be selected from 52 objects without replacement and without regard to order, so N(S) = 52 C 6.

5-20 © 2010 Pearson Prentice Hall. All rights reserved The size of the sample space is EXAMPLE Winning the Lottery Each ticket has two sets of 6 numbers, so a player has two chances of winning for52 6each $1. If E is the event “winning ticket,” then N1E2 = 2 There is about a 1 in 10,000,000 chance of winning the Illinois Lottery!