3. In order to determine insurance premiums, suppose that a car insurance compan

Question

3. In order to determine insurance premiums, suppose that a car insurance company classifies policyholders into one of four classes: excellent risks, good risks, average risks, and bad risks. Their records indicate that the probability an excellent risk individual will be involved in a car accident over a one-year span is 0.02. The same probability for good, average, and bad risk individuals are, respectively, 0.05, 0.14, and 0.32. Of its policy holders, 8% are classified as excellent risks, 16% are classified as good risks, and 62% are classified as average risks. You may assume that car accidents are independent events. (A) What proportion of all policyholders are involved in a car accident within a given one-year period? You must explicitly define all events. (B) If a policyholder did not get into a car accident during 2010, what is the probability they are classified as (i) an excellent risk? (ii) a good risk? (iii) an average risk? (iv) a bad risk?

Explanation / Answer

a) Proportion of policyholders involved in car accident = 0.08*0.02+0.16*0.05+0.62*0.14+0.14*0.32 = 0.1412

b) i) P( Excellent risk/Not Involved in car accident) = (0.08*0.02)/0.1412 = 0.0113

ii) P( good risk/Not Involved in car accident) = (0.16*0.05)/0.1412 = 0.0567

iii) P( average risk/Not Involved in car accident) = (0.62*0.14)/0.1412 = 0.6147

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