To calculate P(H = 1, A = 1), note that

Here, the + operator represents a disjoint union. To get other sorts of probabilities, I think this sort of decomposition is what you need to work with: break up the set of interest into a disjoint union of sets where you are varying over C.

I haven't looked at the problem beyond noticing this, but I'm not terribly optimistic that you can do a whole lot better than this. Maybe you can though.