In a large department store a customer-complaints office handles an average of six complaints per hour about the quality of service. The distribution is Poisson.
a. What is the probability that in any hour exactly six complaints will be received?
b. What is the probability that more than 20 minutes will elapse between successive complaints?
c. What is the probability that fewer than 5 minutes will elapse between successive complaints?
d. The store manager observes the complaints office for a 30-minute period, during which no complaints are received. He concludes that a talk he gave to his staff on the theme "the customer is always right" has obviously had a beneficial effect. Suppose that, in fact, the talk had no effect. What is the probability of the manager observing the office for a period of 30 minutes or longer with no complaints?