Suppose you know how to simulate $X,Y$ two independent random variables in $\mathbb R$ and you are interested in the following quantity : $$ I = \mathbb P \left( \mathbb E \left[ Y|X\right] >a \right), $$ where $a>0$.
Do you know any reference treating non nested lower and upper bounds for this quantity $I$ ? Applying Markov inequality, we would get : $$ I\leq \frac{\mathbb{E}[|X|]}{a} $$ but I am looking at more refined inequalities or even equality with $\sup$ or $\inf$. Thanks.