An auto insurer offers collison coverage to two large groups of policyholders, Group 1 and Group 2. On the basis of historical data, the insurer has determined that the loss due to collision for a policyholder in Group 1 has an exponential distribution with mean 5. On the other hand, the loss due to collision for a policyholder in Group 2 has an exponential distribution with mean 10.
Considering the two groups as one block, about 75% of the losses are from Group 1.
- Given a randomly selected loss in this block, what is the probability that the loss is greater than 15?
- If a randomly selected loss is greater than 15, what is the probability that it is a from a policyholder in Group 1?
An auto insurer has two groups of policyholders – those considered good risks and those considered bad risks. On the basis of historical data, the insurer has determined that the number of car accidents during a policy year for a policyholder classified as good risk follows a binomial distribution with and . The number of car accidents for a policyholder classified as bad risk follows a binomial distribution with and . In this block of policies, 75% are classified as good risks and 25% are classified as bad risks. A new customer, whose risk class is not yet known with certainty, has just purchased a new policy.
- What is the probability that this new policyholder is not accident-free in the upcoming policy year?
- By the end of the policy year, it is found that this policyholder is not accident-free, what is the probability that the policyholder is a “good risk” policyholder?
Soultion is found below.
Solution to Problem 6a
Let be the loss amount of a randomly selected policyholder. The conditional probabilities of a loss greater than 7.5 are:
By the law of total probability, the unconditional probability is:
The above calculation indicates that the unconditional probability is the weighted average of the conditional probabilities. The answer to the second question is obtained by applying the Bayes’ theorem:
Answer to Problem 6b