Just need the 'auto' portion answered. this is more of a general question. 1. The problem statement, all variables and given/known data (This is problem #7 on the sample actuarial exam P, soa 153) An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year and 60% of policyholders who have only a homeowners policy will renew next year. The company estimates that 80% of policyholders who have both an auto and a homeowners policy will renew at least one of those policies next year. Company records show that 65% of policyholders have an auto policy, 50% of policyholders have a homeowners policy, and 15% of policyholders have both an auto and a homeowners policy. Using the company’s estimates, calculate the percentage of policyholders that will renew at least one policy next year. 3. The attempt at a solution My first time looking at this, my thought process was that I will break this up in to cases and sum it all up: Probability they will renew, given that they have Auto only (.5) Probability they will renew, given that they have Homeowners only (.35) Probability they will renew, given that they have Auto and Homeowners (.15) And I immediately though "Bayes theorem." P[A given B] = P [AB]/ P So for Auto, it would be P [ they renewed, and they have auto] / P [they have auto]. which would be .5/.4. When I looked at the solution, it was not .5/.4, but it was .4 * .5. This actually makes sense intuitively (.4 of the .5), but I'm just in general, I dont quite understand when to apply Bayes theorem, and when to not.