I'm not sure I understand what your point is

You set up a certain probability distribution and when you measure you observe the effect of that probability distribution...

Essentially this is the same as sampling from a uniform distribution over the interval [0, 1] and mapping each sample to a discrete value based on cutoff points that split the continuous interval into discrete sections, no?
E.g. like how one can simulate the roll of a die.

So isn't the fact the number of observed discrete values deviates from the theoretical expected counts somewhat... expected?
Same as although probability for each face of a fair six-sided die coming up is 1/6, but when doing 600 actual die rolls not all faces will come up exactly 100 times?