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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 that 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 0.05, 0.14, and 0.32, respectively. 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) Over a given one-year period, what is the probability that all policyholders will be involved in a car accident? Define all events explicitly.
(B) Given that a policyholder was not involved in a car accident during 2010, what is the probability the individual will be classified as (i) an excellent risk? (ii) a good risk? (iii) an average risk? (iv) a bad risk?
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