232 CHAPTER 5 Discrete Probability Distributions 5-2 Beyond the Basics 41. Geometric Distribution If a procedure meets all the conditions of a binomial distribution except that the number of trials is not fixed, then the geometric distribution can be used. The probability of getting the first success on the xth trial is given by P1x2 = p11 - p2x-1, where p is the probability of success on any one trial. Subjects are randomly selected for the National Health and Nutrition Examination Survey conducted by the National Center for Health Statistics, Centers for Disease Control and Prevention. The probability that someone is a universal donor (with group O and type Rh negative blood) is 0.06. Find the probability that the first subject to be a universal blood donor is the fifth person selected. 42. Multinomial Distribution The binomial distribution applies only to cases involving two types of outcomes, whereas the multinomial distribution involves more than two categories. Suppose we have three types of mutually exclusive outcomes denoted by A, B, and C. Let P1A2 = p1, P1B2 = p2, and P1C2 = p3. In n independent trials, the probability of x1 outcomes of type A, x2 outcomes of type B, and x3 outcomes of type C is given by n! 1x12!1x22!1x32! # px1 1 # p x2 2 # p x3 3 A roulette wheel in the Venetian casino in Las Vegas has 18 red slots, 18 black slots, and 2 green slots. If roulette is played 15 times, find the probability of getting 7 red outcomes, 6 black outcomes, and 2 green outcomes. 43.Hypergeometric Distribution If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type (such as lottery numbers you selected), while the remaining B objects are of the other type (such as lottery numbers you didn’t select), and if n objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting x objects of type A and n - x objects of type B is P1x2 = A! 1A - x2!x! # B! 1B - n + x2!1n - x2! , 1A + B2! 1A + B - n2!n! In New Jersey’s Pick 6 lottery game, a bettor selects six numbers from 1 to 49 (without repetition), and a winning six-number combination is later randomly selected. Find the probability of getting exactly four winning numbers with one ticket. Key Concept In Section 5-1 we introduced general discrete probability distributions and in Section 5-2 we considered binomial probability distributions, which is one particular category of discrete probability distributions. In this section we introduce Poisson probability distributions, which are another category of discrete probability distributions. The following definition states that Poisson distributions are used with occurrences of an event over a specified interval, and here are some applications: ■ Number of automobile accidents in a day ■ Number of patients arriving at an emergency room in one hour ■ Number of Internet users logging onto a website in one day 5-3 Poisson Probability Distributions
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