Question

An actuary has done an analysis of all policies that cover two cars. 70% of the policies are of type A for both cars, and 30% of the policies are of type B for both cars. The number of claims on different cars across all policies are mutually independent. The distributions of the number of claims on a car are given in the following table.

Number of Claims	Type A	Type B
0	40%	25%
1	30%	25%
2	20%	25%
3	10%	25%

Four policies are selected at random. Calculate the probability that exactly one of the four policies has the same number of claims on both covered cars. The answer is 0.417. What are the steps to getting this answer?

Answer 1

Clearly, the last step where we select four policies at random, and observe whether the particular policy has the same number of claims on both covered cars, are Bernoulli trials and hence will follow the Binomial distribution. So the final answer is computed by the formula

So the problem boils down to finding the probability of the single event, that we have same no of claims on both cars. It has been explicitly mentioned, that claims on different cars across all policies are mutually independent. This allows us to use the multiplication rule, and establish

(Claims can be same in all 4 cases, when there are wither 0, 1, 2 or 3 claims. In each case, desired probability is obtained by multiplying the respective probability with itself, since we want two cars to have those many no of claims)