An automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume that 70% of the customers are good risks, 20% are medium risks, and 10% are poor risks. Assume that during the course of a year, a good risk customer has probability 0.005 of filing an accident claim, a medium risk customer has probability 0.01, and a poor risk customer has probability 0.025. A customer is chosen at random.

Question

a. What is the probability that the customer is a good risk and has filed a claim?
b. What is the probability that the customer has filed a claim?
c. Given that the customer has filed a claim, what is the probability that the customer is a good risk?

Answer 1

Let G denote the event that a customer is a good risk, let M denote the event that a customer is a medium risk, let P denote the event that a customer is a poor risk, and let C be the event that a customer has filed a claim. Then P(G) = 0.7, P(M) = 0.2, P(P) = 0.1, P(C|G) = 0.005, P(C|M) = 0.01, and P(C|P) = 0.025.



(a) P(G ? C) = P(G)P(C|G) = (0.70)(0.005) = 0.0035



(b) P(C)=P(G ? C) + P(M ? C) + P(P ? C)

=P(G) P(C|G) + P(M) P(C|M) + P(P)P(C|P)

=(0.7)(0.005) + (0.2)(0.01) + (0.1)(0.025)

=0.008