For c you can use the half-Bayes' theorem:An insurance company believes that drivers can be divided into two classes, high-risk and low-risk. According to past data, a high-risk driver has an accident with probability 0.3 during a typical year. On the other hand, a low-risk driver has an accident with probability 0.1 during a typical year. Furthermore, 20% of policyholders are high-risk.
(a) What is the probability that a new policyholder will have an accident within a year of purchasing a policy?
(b) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that that driver is high-risk?
(c) Suppose that a new policyholder has an accident in year one. What is the probability that the policyholder will have an accident in year two? (You may assume that for a given individual, the occurrence of an accident in one year is independent of the occurrence of an accident in another year.)
Using A = "having and accident" and H = "high-risk driver", I found part (a) to be P(A) = 0.14 and part (b) to be P(H|A) = 0.4286 without any problem, but I don't know how to set up part (c). I know I need to take into account that, since the driver had an accident the first year, he may be high-risk. Right now, I'm thinking it could be P(A|H)P(H|A) + P(A|H')P(H'|A), but I'm not sure.
Here denotes an accident in year 2 and an accident in year 1.