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Counting Techniques in Probability

Learn how to solve counting problems using the multiplication rule, permutations, and combinations. Discover how to compute probabilities involving permutations and combinations.

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Counting Techniques in Probability

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  1. Chapter 5 Probability

  2. Section 5.5 Counting Techniques

  3. Objectives • Solve counting problems using the Multiplication rule • Solve counting problems using permutations • Solve counting problems using combinations • Solve counting problems involving permutations with nondistinct items • Compute probabilities involving permutations and combinations

  4. Objective 1 • Solve Counting Problems Using the Multiplication Rule

  5. Multiplication Rule of Counting If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in different ways.

  6. EXAMPLE Counting the Number of Possible Meals For each choice of appetizer, we have 4 choices of entrée, and for each of these 2 • 4 = 8 parings, there are 2 choices for dessert. A total of 2 • 4 • 2 = 16 different meals can be ordered.

  7. If n ≥ 0 is an integer, the factorial symbol, n!, is defined as follows: n! = n(n – 1) • • • • • 3 • 2 • 1 0! = 1 1! = 1

  8. Objective 2 • Solve Counting Problems using Permutations

  9. A permutation is an ordered arrangement in which r objects are chosen from n distinct (different) objects and repetition is not allowed. The symbol nPr represents the number of permutations of r objects selected from n objects.

  10. Number of permutations of n Distinct Objects Taken r at a Time • The number of arrangements of r objects chosen from n objects, in which • the n objects are distinct, • repetition of objects is not allowed, and • order is important, is given by the formula

  11. EXAMPLE Betting on the Trifecta In how many ways can horses in a 10-horse race finish first, second, and third? 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

  12. Objective 3 • Solve Counting Problems Using Combinations

  13. A combination is a collection, without regard to order, of n distinct objects without repetition. The symbol nCr represents the number of combinations of n distinct objects taken r at a time.

  14. Number of Combinations of n Distinct Objects Taken r at a Time • The number of different arrangements of r objects chosen from n objects, in which • the n objects are distinct, • repetition of objects is not allowed, and • order is not important, is given by the formula

  15. EXAMPLE Simple Random Samples How many different simple random samples of size 4 can be obtained from a population whose size is 20? 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.

  16. Objective 4 • Solve Counting Problems Involving Permutations with Nondistinct Items

  17. Permutations with Nondistinct Items The number of permutations of n objects of which n1 are of one kind, n2are of a second kind, . . . , and nk are of a kth kind is given by where n = n1 + n2 + … + nk.

  18. EXAMPLE Arranging Flags How many different vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red? 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

  19. Objective 5 • Compute Probabilities Involving Permutations and Combinations

  20. EXAMPLE Winning the Lottery 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?

  21. 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) = 52C6.

  22. EXAMPLE Winning the Lottery The size of the sample space is Each ticket has two sets of 6 numbers, so a player has two chances of winning for each $1. If E is the event “winning ticket,” then There is about a 1 in 10,000,000 chance of winning the Illinois Lottery!

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