Not sure I phrase the title right, but here's by question in a nutshell.
Say you have a box of balls with different colors. Let's say there are $N$ balls of $k$ different colors, and that the distribution of colors is such that some colors are more represented than others with a known distribution. You then ask a completely colorblind person to separate the balls across $M$ equally probable bins.
What is the likelihood of the distribution of balls made by this colorblind person?
Or: how likely is it that the person is really colorblind?
So far I've addressed the problem in this line of thought: If the person is completely colorblind, the most likely distribution would be random so it makes sense to somehow compare the outcome to this. If there were infinite balls I would just compare the distribution in each bin to the known distribution of colors, but as $N$ grows smaller this approach reduces in validity. The problem can also be recast such as to say that each bin draws a number of times from the distribution, but it gets very complicated when $N$ is finite because then the draws in each bin influence each other. Not sure how to proceed from here.
Disclaimer: This is a simplification of the actual problem I'm working on. Really, I'm studying whether the values a variable assumes in varying distances from a point can be said to have statistical significance, or are just random. The $N$ values of the variable are discretised into $k$ intervals, and distances into $M$ intervals.