Suppose $X$ is discrete and $Z,W$ are discrete or continuous, I am wondering if it is always the case (or at least non-trivially) that
$$ P(X=x\mid Z) \geq P(X=x\mid Z,W) $$
for all $x\in X$.
It appears to make intuitive sense to me because it appears the probability of $X$ given $Z$ and $W$ should be "subsetting" off the probability of $X$ given $Z$. That is, if there is some chance of $X$ given $Z$, then $X$ given $Z$ and $W$ is subdividing the occurrence of $X$ given $Z$ into many more chunks according to $W$, and thus the probability of a chunk occurring should be less than the whole.
In other words, it seems to hold in the finite-sampling perspective, where the above are empirical proportions over some collection of objects. However, the result doesn't seem to hold more generally. I am wondering why the finite-sampling intuition doesn't seem to extend. Is there a general result behind this?