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.