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.
Best Answer
Your reasoning is correct that $A$ and $B$ are dependent. At the same time, your intuition that it is hard to work out probabilities without exploiting some form of independence is correct as well. The next best thing to independence is conditional independence, which is what this exercise is teaching you to use. The idea is that when we lump good and bad drivers together we have lost the independence (since good vs bad is a confounding variable) but by splitting them apart, we can restore the independence.
Using this "splitting out" philosophy, our way of calculating $P(G|A^c)$ is to expand it out as $$ P(G|A^c)=\frac{P(G\cap A^c)}{P(G\cap A^c)+P(G^c\cap A^c)}=\frac{g(1-p_1)}{g(1-p_1)+(1-g)(1-p_2)}, $$ and we can use the same approach in a slightly more elaborate way to solve part (c) - you will need to write both the numerator and denominator as a sum over cases depending on whether the driver is good or bad, using conditional independence to compute each term in the sum as a product of the corresponding probabilities.
One final question: do you see why the answer to (a) relies on the assumption $p_1\not=p_2$?