Expectation E(Y|X) when P(Y|X) is binomial

Question

Suppose $X \sim Bin(n, p_1)$, and $Y|X \sim Bin(m, p_2)$, and I want to calculate E(Y|X).

I can start this off:

$E(Y|X) = \sum_x xP(Y|X)$

$= \sum_x x {x \choose y} p_2^y (1-p_2)^{(x-y)}$

but now what?

Answer 1

If $Y|X \sim Bin(m, p_2)$, then $E(Y|X)=mp_2$. The fact it is a conditional statement is irrelevant.