744 CHAPTER 11 Probability Suppose that a basket contains three identical balls, except for their color. One is red, one is blue, and one is yellow (Fig. 11.21). Suppose further that we are going to randomly select three balls with replacement from the basket. We can determine specific probabilities by examining the tree diagram shown in Fig. 11.22. Note that 27 different selections are possible, as indicated in the sample space. Sample Space rrr rry rrb rbr rby rbb ryr ryy ryb brr bry brb bbr bby bbb byr byy byb yrr yry yrb ybr yby ybb yyr yyy yyb r b y r b y r b y r b y r b y r b y r b y r b y r b y r b y y b r y b r y b r Figure 11.22 Suppose that you are a server at a restaurant and have learned from past experience that 80% of your customers leave a tip. If you wait on 6 customers, what is the probability that all 6 customers will leave a tip? If you wait on 8 customers, what is the probability that at least 5 of them will leave a tip? In this section, we will learn how to use the binomial probability formula to answer these and similar questions. Why This Is Important Many applications from business and industry involve the use of binomial probability. Insurance companies use binomial probability when setting insurance rates. Pharmacological companies use binomial probability to assess the effectiveness of their drugs. Binomial probability also plays a vital role in other branches of mathematics—especially statistics—and in computer science, biology, chemistry, and physics. Binomial Probability Formula SECTION 11.10 LEARNING GOAL Upon completion of this section, you will be able to: 7 Solve problems using the binomial probability formula. Figure 11.21 Research Activity 42. Poker Probabilities Throughout this section we have discussed the probabilities of several poker hands. Do research and write a paper on how to calculate the probabilities for all poker hands. Include in your report one pair, two pairs, three of a kind, straight, flush, full house, four of a kind, straight flush, and royal flush. Recreational Mathematics 41. Hair When the Isle of Flume took its most recent census, the population was 100,002 people. Everyone on the island has fewer than 100,001 hairs on his or her head. Determine the probability that at least two people have exactly the same number of hairs on their head. 1. The options for the number of hairs are 0, 1, 2, 3, . . . , 100,000. Since there are more people than options, two or more people must have the same number of hairs on their head. StatCrunch Applets Simulation Urn Sampling Paul Vasarhelyi/Shutterstock
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