How to prove $X$~$B(n,p)$ and $Y$~$B(m,p)$?

Question

Assuming $X$ and $Y$ are random variables with non-negative integer values, and they are independent. $\forall x,y \in N^+$, we have $P(X=x| \ X+Y=x+y)=\frac{\binom{m}{x}\binom{n}{y}}{\binom{m+n}{x+y}}$, which m,n is positive integer. And $P(X=0)>0, P(Y=0)>0$.

So how to prove $X$~$B(n,p)$ and $Y$~$B(m,p)$ now?

Both attempts have failed and I'm not quite sure what else to try. Any help would vastly appreciated.

Answer 1

Hint Use the formula which says that if random variables $X$ and $Y$ are independent, then $f_{X|Y=y}(x,y)=\frac{f_{XY}(x,y)}{f_Y(y)}.$