If n = 4900, what is the probability that at least 2 chips are in a randomly If n = 4900, what is the number of cookies in each batch that are expected to FDA regulations declare that at most 1 of cookies labeled “chocolate chip” The shop decides to add f

2.14 The Poisson Process

169 a. What is the probability that at most 2 books are misshelved? b. Approximate the probability in part a using the Poisson distribution. 17. A popular chocolate chip cookie “guarantees” at least 16 chocolate chips per cookie. The actual number of chocolate chips per cookie, however, is a Pois- son random variable. What must be the average number of chips per cookie if approximately 95 or more of the cookies are to meet the guarantee?

18. A bakery makes a batch of 1000 chocolate chip cookies and adds n chocolate chips

to the batter for each batch and mixes the batter well. Under these assumptions, the number of chocolate chips per cookie should follow a Poisson distribution.

a. If n = 4900, what is the probability that at least 2 chips are in a randomly

selected cookie?

b. If n = 4900, what is the number of cookies in each batch that are expected to

contain exactly 3 chocolate chips?

c. FDA regulations declare that at most 1 of cookies labeled “chocolate chip”

can fail to contain a single chocolate chip. What is the minimum value for n for the bakery to be within the law?

19. A truck repair shop has facilities for the repair of 3 large trucks per day. The trucks

arrive according to a Poisson process with 2 trucks expected per day. If more than 3 trucks arrive, the excess is turned away. a. Find the probability that exactly 3 trucks arrive in one day. b. Find the probability that trucks are turned away. c. Find the probability distribution for X, the number of trucks serviced per day. d. Find the expected number of trucks turned away each day.

e. The shop decides to add facilities so that it can service the trucks arriving

during a day about 95 of the time. How many trucks must it be able to service in a day?

20. Calls come into a very busy switchboard at a rate of 6 per minute according to

a Poisson process. Unfortunately, some new electronic switching devices work imperfectly and the probability that a received call is switched to the proper extension is only 0.8. It has been observed that the calls are switched independently, however.

a. If X represents the number of calls correctly switched, find P. X = k for

some one-minute period. b. Simplify the result in part a and show that X is Poisson with parameter 4.8. 21. Telephone calls coming into a busy switchboard follow a Poisson distribution with 4 calls expected in a one-minute period. The switchboard, however, can answer at most 6 calls in a one-minute interval; any calls exceeding 6 during that period receive a busy signal. 170

Chapter 2 Discrete Random Variables and Probability Distributions