234 CHAPTER 5 Discrete Probability Distributions Poisson Distribution as Approximation to Binomial The Poisson distribution is sometimes used to approximate the binomial distribution when n is large and p is small. One rule of thumb is to approximate the binomial distribution by using the Poisson distribution when the following two requirements are both satisfied. Requirements for Using Poisson as an Approximation to Binomial 1. n Ú 100 2. np … 10 If both requirements are satisfied and we want to use the Poisson distribution as an approximation to the binomial distribution, we need a value for m. That value can be calculated by using Formula 5-6 (from Section 5-2): c. In this 118-year period, there were actually 16 years with 6 Atlantic hurricanes. How does this actual result compare to the probability found in part (b)? Does the Poisson distribution appear to be a good model in this case? a. The Poisson distribution applies because we are dealing with the occurrences of an event (hurricanes) over some interval (a year). The mean number of hurricanes per year is m = number of hurricanes number of years = 652 118 = 5.5 b. Using Formula 5-9, the probability of x = 6 hurricanes in a year is as follows (with x = 6, m = 5.5, and e = 2.71828): P162 = mx # e-m x! = 5.56 # 2.71828-5.5 6! = 127,680.64210.00408678662 720 = 0.157 The probability of exactly 6 hurricanes in a year is P162 = 0.157. c. The probability of P162 = 0.157 from part (b) is the likelihood of getting 6 Atlantic hurricanes in 1 year. In 118 years, the expected number of years with 6 Atlantic hurricanes is 118 * 0.157 = 18.5 years. The expected number of years with 6 hurricanes is 18.5, which is reasonably close to the 16 years that actually had 6 hurricanes, so in this case, the Poisson model appears to work quite well. SOLUTION YOUR TURN. Do Exercise 5 “Hurricanes.” FORMULA 5-6 Mean for Poisson as an Approximation to Binomial m = np Maine Pick 4 EXAMPLE 2 In the Maine Pick 4 game, you pay 50. to select a sequence of four digits (0–9) with repetition allowed, such as 1377. If you play this game once every day, find the probability of winning at least once in a year with 365 days.
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