The Law of Iterated Expectations works for random variables $X$ and $Y$ as $E_{Y}[E[X|Y]] = E_X[X]$. However, if instead of $E[X|Y]$ we take $Var(X|Y)$, i.e. conditional variance, then we know that $E_{Y}[Var(X|Y)]$ $\neq$ $E_X[Var(X)]=Var(X)$. Consider an expression $f(X,Y)$, which is a general notation for $E[X|Y]$ or $Var(X|Y)$ or something else related to $X$ and $Y$.
The question is what conditions we have to impose on this expression $f(X,Y)$ and some other expression $g(X)$, for the following to be true ( or not to be true): $E_{Y}[f(X,Y)]=E_X[g(X)]$, or may be $E_{Y}[Pr[f(X,Y)<0|Y]]=Pr[f(X,Y)<0]$. Thank you