Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$.
My question: is there an explicit formula for the variance of $X_1$ conditioned on $S=s$? If it makes it any easier, in my application $s=\frac{mnN}{M}$, which is the mean.
I know how to get approximations that are adequate for my purposes, but still I wouldn't want to miss a useful exact formula if one is known.
