I am working on a problem from the book "probability and random process" by Geoffrey Grimmett.
suppose X, Y are independent random variables take values of non-negative integers. and they have the following property. $$P\left(X=k|X+Y=n\right)=\left(\begin{array}{c} n\\ k \end{array}\right)p^{k}\left(1-p\right)^{n-k} $$ which is a binomial distribution, prove X, and Y are poisson random variables.
I know the result that "conditioning on X+Y, X or Y obeys binomial distribution" from the property of poisson process. However, I don't know how to prove from formula to poisson.
Any help will be appreciated, thanks.