A car insurance company has high-risk, medium-risk, and low-risk clients, who have, respectively, $0.04, 0.02,$ and $0.01$ of filing claims within a given year. The proportions of the numbers of clients in the three categories are $0.15, 0.25,$ and $0.60,$ respectively.
(a) What is the probability that a random client doesn't file a claim?
(b) What proportion of the claims filed each year come from high-risk clients?
(c) What is the probability that a random client who didn't file a claim is low-risk?
I definitely think I can solve this problem. I think my issue is that I'm having trouble setting up the variables to work with. The probabilities and relevant info are obviously given in the question. It's just that setting it all up is a weak point of mine.
Best Answer
Let $H,M,L$ be the event for a customer is in the high-, medium-, low- risk brackets respectively (disjoint and exhaustive). Let $F$ be the event for a customer filing a claim in the year.
$${\mathsf P(F\mid H)=0.04\\\mathsf P(F\mid M)=0.02\\\mathsf P(F\mid L)=0.01}\qquad{\mathsf P(H)=0.15\\\mathsf P(M)=0.25\\\mathsf P(L)=0.60}$$
So, yes indeed, you are clearly expected to use Bayes' Rule and the Law of Total Probability to find the answers. Can you now do that?