I have two events at my disposal where the objective is to find out which of the two is more likely to occur. After some manipulation I get:
$$ \Pr\left( A\geq \theta \mid A\leq B \right) - \Pr\left( A\geq \theta \mid A> B \right) $$
where $A$ and $B$ are random variables of some given distributions, and $\theta>0$ is a constant.
Computing these probabilities is not straightforward since we are conditioning on an event rather than a single random variable.
Nevertheless, based on the form of the expression, it is easy to see that the difference is $<0$. However, I get contradicted by my simulation results which indicate that the difference is not necessarily less than zero.
I believe my simulations are in order, leading me to believe that there is instead something wrong in the above reasoning?