An automobile insurance company divides customers into three categories, good ri

Question

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

Explanation / Answer

Let

G = good risk
M= medium risk
P = poor risk
A = accident

a)

P(G n A) = P(G) P(A|G) = 0.70*0.005 = 0.0035 [ANSWER]

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b)

By Bayes' RUle,

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

= 0.70*0.005 + 0.20*0.01 + 0.10*0.025

= 0.008 [ANSWER]

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c)

P(G|A) = P(G) P(A|G) / P(A) = 0.70*0.005/0.008 = 0.4375 [ANSWER]

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