Conditional Expectation and Conditional Varience

Question

Suppose that X|Y = p has a Binomial(n,p) distribution for 0 < p <1 and Y has a U(0,1) distribution. Find E(X) and Var(X).

Answer 1

$\mathsf EX=\mathsf E(\mathsf E(X\mid Y))=\mathsf EnY=n\mathsf EY=\frac12n$.

You can find $\mathsf{Var}X$ on base of $\mathsf{Var}(X)=\mathsf EX^2-(\mathsf EX)^2$ and:

$\mathsf EX^2=\mathsf E(\mathsf E(X^2\mid Y))=\dots$

I leave the rest to you.