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?