SECTION 4.2 Binomial Distributions 213 Unusual Events In Exercises 37 and 38, find the indicated probabilities. Then determine if the event is unusual. Explain your reasoning. 37. Rock-Paper-Scissors The probability of winning a game of rock-paper-scissors is 1 3.You play nine games of rock-paper-scissors. Find the probability that the number of games you win is (a) exactly five, (b) more than five, and (c) less than two. 38. Marriage Fifty-three percent of U.S. adults are currently married. You randomly select twelve U.S. adults. Find the probability that the number who are married is (a) exactly nine, (b) less than four, and (c) from eight to eleven. (Source: Pew Research) Extending Concepts Multinomial Experiments In Exercises 39 and 40, use the information below. A multinomial experiment satisfies these conditions. • The experiment has a fixed number of trials n, where each trial is independent of the other trials. • Each trial has k possible mutually exclusive outcomes: E1, E2, E3, . . ., Ek . • Each outcome has a fixed probability. So, P1E12 = p1, P1E22 = p2, P1E32 = p3, . . ., P1Ek2 = pk. The sum of the probabilities for all outcomes is p1 + p2 + p3 + g+ pk = 1. • The number of times E1 occurs is x1, the number of times E2 occurs is x2, the number of times E3 occurs is x3, and so on. • The discrete random variable x counts the number of times x1, x2, x3, . . ., xk that each outcome occurs in n independent trials where x1 + x2 + x3 + g+ xk = n. The probability that x will occur is P1x2 = n! x1!x2!x3! gxk! p1 x1p 2 x2p 3 x3 gpk xk. 39. Genetics According to a theory in genetics, when tall and colorful plants are crossed with short and colorless plants, four types of plants will result: tall and colorful, tall and colorless, short and colorful, and short and colorless, with corresponding probabilities of 9 16, 3 16, 3 16, and 1 16. Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless. 40. Genetics Another proposed theory in genetics gives the corresponding probabilities for the four types of plants described in Exercise 39 as 5 16, 4 16, 1 16, and 6 16. Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless. 41. Manufacturing An assembly line produces 10,000 automobile parts. Twenty percent of the parts are defective. An inspector randomly selects 10 of the parts. (a) Use the Multiplication Rule (discussed in Section 3.2) to find the probability that none of the selected parts are defective. (Note that the events are dependent.) (b) Because the sample is only 0.1% of the population, treat the events as independent and use the binomial probability formula to approximate the probability that none of the selected parts are defective. (c) Compare the results of parts (a) and (b).
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