I'm working on this below problem, and am having difficulty.
Suppose that there are two types of drivers: good drivers and bad drivers. Let G be the event that a certain man is a good driver, A be the event that he gets into a car accident next year, and B be the event that he gets into a car accident the following year. Let $P(G) = g$ and $P(A|G) = P(B|G) = p_1$, $P(A|G^c) = P(B|G^c) = p2$, with $p1 < p2$. Suppose that given the information of whether or not the man is a good driver, A and B are independent (for simplicity and to avoid being morbid, assume that the accidents being considered are minor and wouldn’t make the man unable to drive).
(a) Explain intuitively whether or not A and B are independent.
(b) Find $P(G|A^c)$.
(c) Find $P(B|A^c)$.
It seemed to me that $A$ and $B$ were not independent, since if a driver gets into an accident in year $i$, he is more likely to fit into the "bad driver" category, and so will be more likely to get into an accident in year $i + 1$. The event $A$ will then increase the probability of the event $B$, in which case they wouldn't be unconditionally independent.
However, this creates quite a lot of issues in terms of working out part (c) due to the absence of information of conditional probabilities involving $B$ and $A$, which seems to almost imply that $A$ and $B$ must be independent, which would mean that $B$ and $A^c$ are also independent. But even though I think I have worked backwards to the intended answer, I am having difficult understanding why this is the case.
I would greatly appreciate any helpful insights.