task with pousson distribution

Hello!

I am stuck on one task:

An insurance company divides its customers into 2 groups. Twenty percent

of customers are in the high-risk group, and eighty percent are in the low-risk

group. The high-risk customers make an average of 1 accident per year while

the low-risk customers make an average of 0.1 accidents per year. Mr. X had

no accidents last year. What is the probability that he is a high-risk driver?

I made something like this:

A - event Mr. X is high-risk driver

B - event Mr. X is low-risk driver

Y - random variable of number of times Mr.X had accidents last year.

By Poisson I got

P (Y=0 | A ) = 0,368 (lambda=1)

P(Y=0 | B ) = 0,905 (lambda=0.1)

So I guess I need to find

P(A | Y=0) = ? or maybe P (A ) = ?

There are two more parameters 0.2 probability of being high-risk driver and 0.8 probability of being low-risk driver.

Please help!

Thanks in advance.