My work:
For part a) I have $0.14$ x $0.04= 0.0004$
For part b) I'm confused between the following two calculations. Would it be $0.055 + 0.023 + 0.04$ or would it be $0.16$ x $0.055 + 0.70$ x $0.023 + 0.14$ x $0.04$?
For part c) I have $P(Over 65 | Accident)$= $\frac{answer from part (a)}{answer from part (b)}$
Can someone please clarify these? Thanks!
Your approach to (a) is correct but your computation is not. $$0.14 \times 0.04 = 0.0056.$$ I have no idea how you got $0.0004$.
I should point out that while part (a) asks "What is the probability...accident and the driver was over 65," the intent should be "What is the probability...accident and the driver was 65 or over." The reason is that the last category of drivers includes ages 65 and over. Otherwise, you would have insufficient information to answer the question as originally stated.
For part (b), your first calculation is incorrect because it does not weight the probability of accident by the proportion of drivers in the respective age group. Therefore, the second calculation is the correct one.
For part (c), your approach is correct but you should explain your reasoning formally. Let $B$ be the event a randomly selected driver is at least age $65$. Let $A$ be the event that a randomly selected driver had an accident. Then you are asked to compute $\Pr[B \mid A]$, the probability that, given a randomly selected driver had an accident, that this driver was at least age $65$. By definition of conditional probability, $$\Pr[B \mid A] = \frac{\Pr[B \cap A]}{\Pr[A]}.$$ The numerator is the joint probability that a randomly selected driver had an accident and is at least age $65$, which is the computation from part (a). The denominator is the unconditional/marginal probability of a randomly selected driver having an accident, which is the computation from part (b).