I have a question that I'm having some trouble with, but which I believe might have a fairly straightforward answer. I'd really appreciate it if someone could help point me in the right direction!
Let's say I have three binomial random variables, $C\sim\text{B}(N,c)$, and two which depend on it via $H\sim\text{B}(C,\gamma)$, and $N\sim\text{B}(C,\alpha)$, where $\gamma+\alpha\leq 1$.
What I'm looking for is the form of the conditional distribution $P(N\mid H)$. If it helps, I'm specifically interested in the case where $P(N\mid H=0)$.
Edit: I additionally require that $H+N\leq C$, which is the part that I'm not entirely sure how to deal with. If there's a better way to ask the question, I would really appreciate being corrected. I'm trying to model a process which occurs in two steps. In the first step, a number of events $C$ occur according to $C\sim B(N,c)$. Then in the second step, each one of those events can be classified either as $N$ or $H$, which is where the other two binomials came from. Perhaps this would be better modeled as a multinomial?
Thanks in advance!