SECTION 4.3 More Discrete Probability Distributions 217 TRY IT YOURSELF 1 The study in Example 1 found that the failure rate of businesses after one year is 20%. Six businesses that started one year ago are selected at random. Find the probability that the sixth business selected is the first one to have failed. (Source: U.S. Bureau of Labor Statistics) Answer: Page A39 Even though theoretically a success may never occur, the geometric distribution is a discrete probability distribution because the values of x can be listed: 1, 2, 3, . . .. Notice that as x becomes larger, P1x2 gets closer to zero. For instance, in Example 1, the probability that the thirtieth business selected is the first one to have failed is P1302 = 0.5010.50230-1 = 10.50230 ≈ 0.0000000009. The Poisson Distribution In a binomial experiment, you are interested in finding the probability of a specific number of successes in a given number of trials. Suppose instead that you want to know the probability that a specific number of occurrences takes place within a given unit of time, area, or volume. For instance, to determine the probability that an employee will take 15 sick days within a year, you can use the Poisson distribution. The Poisson distribution is a discrete probability distribution of a random variable x that satisfies these conditions. 1. The experiment consists of counting the number of times x an event occurs in a given interval. The interval can be an interval of time, area, or volume. 2. The probability of the event occurring is the same for each interval. 3. The number of occurrences in one interval is independent of the number of occurrences in other intervals. The probability of exactly x occurrences in an interval is P1x2 = m xe-m x! where e is an irrational number approximately equal to 2.71828 and m is the mean number of occurrences per interval unit. DEFINITION Using the Poisson Distribution The mean number of accidents per month at a certain intersection is three. What is the probability that in any given month four accidents will occur at this intersection? SOLUTION Using x = 4 and m = 3, the probability that 4 accidents will occur in any given month at the intersection is P142 ≈ 3412.718282-3 4! ≈ 0.168. You can use technology to check this result. For instance, using a TI-84 Plus, you can find P142, as shown at the left. EXAMPLE 2 TI-84 PLUS poissonpdf(3,4) .1680313557 Tech Tip You can use technology such as Minitab, Excel, StatCrunch, or the TI-84 Plus to find a Poisson probability. For instance, here are instructions for finding a Poisson probability on a TI-84 Plus. From the DISTR menu, choose the poissonpdf( feature. Enter the values of m and x. (Note that the TI-84 Plus uses the Greek letter lambda, l, in place of m.2 Then calculate the probability.
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