I am looking too see that $\mathbb{P}(X > \mathbb{E}[X|\mathscr{F}])|\mathscr{F})$ is not the same as $\mathbb{P}(X > \mathbb{E}[X])$$.
In other words, some event happens and I want to calculate the probability of the conditional expectation given the event, as opposed to given no such event.
I don't think this is true, but I want to make sure I am not repeating the same work I've done.
If $X$ is measurable w.r.t. $\mathcal F$ then $E(X|\mathcal F)=X$ so the left side is $P(X>X|\mathcal F)=0$ almost surely. $P(X>EX)$ can be any number between $0$ and $1$ (It is $\frac 1 2$ if $X$ has standard normal distribution).