3 category car insurance probability

I couldn’t find any 3 category car insurance examples. I was able to deduce part a but I could use some help trying to figure out part b) of the following:

An insurance company believes that people can be classified into three groups: good risk,
average risk or bad risk. Their statistics show that the probabilities of good, average and bad
risk individuals being involved in an accident in any one year are $0.04$, $0.12$ and $0.3$ respectively.
Assume that $20$% of the population can be classified as good risk, $55$% as average risk and $25$% as
bad risk.

a. Find the proportion of policy holders having accidents in any one year.

b. Suppose that a new policy holder has an accident within a year of purchasing a policy. Find
the probability that the policy holder is an average risk.

For part a, I think it was just asking for the total probability of an accident which was P(A) = P(good)P(A|good) + P(avg)P(A|avg) + P(bad)P(A|bad), which is: $$.2(.04)+.55(.12)+.25(.3) = .149.$$

I think part b) involves Bayes’ theorem where P(avg|A) = (P(A|avg)P(avg))/P(A), and where the denominator is the answer from part a, and the numerator are given parts from the statement.

Thanks in advance! 🙂