Conditional Probabilty question

okay the problem statement is as follows

A gene can be either type A or type B (ie note type A). A gene can also be either dominant or regressive.

If a gene is type B, then there is a 0.31 probability that it is dominant. When a gene is dominant, then there is also a 0.22 probability that it is type B.

Let A be the event that the gene is of 'type A' and let D be the even that the gene is 'dominant'. What is the probability that the gene is type A?

so what is given is P(dominant|B)=0.31 and P(B|dominant)=0.22 and that A and B are mutually exclusive

Now what I'm not entirely sure is correct is, P(regressive|B)=0.69 and P(A|dominant)=0.78

Breaking down P(dominant|B) = P(dom intersect B)/P(B) and similarly for the others. I've tried every combination I can think of for solving for P(A) and I always end up with more unknowns than equations. What should I do?

I also have P(A) = (A int R + A int D)/(A int R + A int D + B int R + B int D)

similarly for P(B) = (B int R + B int D)/(A int R + A int D + B int R + B int D)