## If $X$ is discrete and $Z,W$ are discrete or continuous, is it always the case that $P(X=x\mid Z) \geq P(X=x\mid Z,W)$?

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?