An insurance company found that 2.5% of male drivers between the ages of 18 and 25 are involved in serious accidents annually. To simplify the analysis, assume that every such accident costs the insurance company $65,000 and that a driver can only have one of these accidents in a year.
(a) If the company charges $2,500 for such coverage, what is the chance that it loses money on a single policy?
(b) Suppose that the company writes 1,000 such policies to a collection of drivers. What is the probability that the company loses money on these policies? Assume that the drivers don’t run into each other and behave independently.
(c) Does the difference between the probabilities of parts (a) and (b) explain how insurance companies stay in business? Large auto insurers are certainly profitable.