probability of Conditional Expectation

Question

Is there a general formula for this type of questions?

Given data: $$X \sim \mathrm{Geo}(0.09)$$ $$Y|X=x \sim \mathrm{Geo}(1/x+1)$$

How do I calculate $\mathbb P(\mathbb E(Y|X)=3)$?

Answer 1

First determine what is $\mathbb E(Y\mid X)$.   (Well, it is a function of $X$, but what function is it?)

Let $g(x)=\mathbb E(Y\mid X=x)$. Evaluate $\mathbb P(g(X)=3)$