I don't remember whether it is likely to work to consider the
probability of each population being correct given the content of each
individual bin. I think i've tried this a couple times, but I'm not
sure my work was sound. I'm trying to understand the theory of it.
Is it reasonable to count the number of ways a bin could be sampled
from each population, and divide? If this is reasonable, it may give
the most accurate answer. It may even be easy to do. If it involves
factorial, most of them may cancel when placed in ratio, or there may
be a distribution that gives good results when the sample count is
large.