as usual this is incompete, are you conditioning on X,Y, X+Y....?

Since the binomial is finite and the Poissons are infinite, my guess is that you want X|X+Y or Y|X+Y

since these will be bounded

In any case use P(A|B)=P(AB)/P(B).

AND X+Y is Poisson.

A similar example can be found at the end of....http://mathworld.wolfram.com/BinomialDistribution.html

DO exactly what they did but at eq (66) use the Poisson and what I said about the sum of two indep Poissons.