11.9 Solving Probability Problems by Using Combinations 741 b) In Section 11.4, we introduced the formula: P P (event happening at least once) 1 (event does not happen). = − Applying this formula to this problem we have: P Toy Story P Toy Story (at least one movie) 1 (no movies). = − In part a), we determined that P(no Toy Story movies) 1 6 = . Therefore, P Toy Story (at least one movie) 1 1 6 5 6 = − = c) The number of ways to select one Toy Story movie is C4 1. The number of ways to select one Cars movie is C3 1. And the number of ways to select one Monsters, Inc. movie is also C3 1. The total number of movie selections is C 10 3 Therefore, P C C C C (one movie from each franchise) 4 3 3 120 36 120 3 10 4 1 3 1 3 1 10 3 = ⋅ ⋅ = ⋅ ⋅ = = 7 Now try Exercise 27 Example 4 Rare Coins Conner’s rare coin collection is made up of 8 silver dollars, 7 quarters, and 5 dimes. Conner plans to sell 8 of his 20 coins to finance part of his college education. If he selects the coins at random, what is the probability that 3 silver dollars, 2 quarters, and 3 dimes are selected? Solution The number of ways that Conner can select 3 out of 8 silver dollars is C . 8 3 The number of ways he can select 2 out of 7 quarters is C . 7 2 The number of ways he can select 3 out of 5 dimes is C . 5 3 He will select 8 coins from a total of 20 coins. The number of ways he can do so is C . 20 8 The probability that Conner selects 3 silver dollars, 2 quarters, and 3 dimes is calculated as follows. P C C C C (3 silver dollars, 2 quarters, and 3 dimes) 56 21 10 125,970 11,760 125,970 392 4199 8 3 7 2 5 3 20 8 = ⋅ ⋅ = ⋅ ⋅ = = 7 Now try Exercise 23 Learning Catalytics Keyword: Angel-SOM-11.9 (See Preface for additional details.) Instructor Resources for Section 11.9 in MyLab Math • Objective-Level Videos 11.9 • Interactive Concept Video: Sample Spaces • StatCrunch: Simulation: Poker Hands • PowerPoint Lecture Slides 11.9 • MyLab Exercises and Assignments 11.9 Exercises Practice the Skills In Exercises 1–8, set up the problem as if it were to be solved, but do not solve. Assume that each problem is to be done without replacement. 1. Pens A bin full of promotional pens for the University of Florida contains 12 blue pens and 8 orange pens. If Lindsay randomly selects 4 pens from the bin, determine the probability that all 4 pens are blue. C C 12 4 20 4 2. Stock Car Race The Crawford County Speedway held a race with 18 Ford cars and 12 Chevrolet cars. If 8 of these cars are randomly selected, determine the probability that all 8 are Chevrolet cars. C C 12 8 30 8 m See Exercise 2 3. Kings Determine the probability of being dealt 3 kings from a standard 52-card deck when 3 cards are dealt. C C 4 3 52 3 4. Letters Three letters are to be randomly selected from 26 letters. Determine the probability that 3 vowels out of the 5 vowels (a, e, i, o, u) are selected. C C 5 3 26 3 actionsports/123RF SECTION 11.9 Applets Simulation Poker Hands StatCrunch
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