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.