Asked this question over at the stats forum and did not get a reply: https://stats.stackexchange.com/questions/169572/test-of-strict-difference-in-independent-binomial-probabilities
Suppose $X,Y$ are two independent binomial random variables with parameters $n_1,p_1$ and $n_2,p_2$ respectively. Suppose one wanted to test the hypothesis $p_1>p_2$. Conditional on $X+Y=s$, would it still be valid to construct a critical value $x_1$ based on the hypergeometric distribution which assumes equality of $p_1$ and $p_2$ as in Fisher's exact test and then reject the hypothesis if $X\leq x_1$? Intuitively this seems like it would be ok, but since the distribution of $X$ conditional on $X+Y=s$ is not hypergeometric if $p_1\neq p_2$ I was thinking there may be some issues with this procedure.