I have got stuck with a problem related to the multivariate hypergeometric distribution. I pick $n$ balls from an urn containing a total of $N$ balls. Each ball is painted with (possibly multiple) coloured dots. Eventually I'd like to be able to deal with $p$ possible colours, but for the time being say $p=3$ and that the colours are red, green and blue. I have a big table with the total number of balls with each of the $2^p$ dot combinations.

I want to compute the joint probability for ending up with $k_r$ balls with a red dot, $k_g$ balls with a green dot and $k_b$ balls with a blue dot. To me it looks like the multivariate hypergeometric distribution but with overlapping ball categories. Has anyone seen this problem before? Could you please point me in a promising direction?

  • $\begingroup$ Did you at least take a look at the wiki section? Not to mention all the other Google hits. $\endgroup$ – Lee David Chung Lin Feb 25 at 8:38
  • $\begingroup$ @LeeDavidChungLin Thanks for the suggestion. Yes, I did. I can only find examples when each ball belongs to a single colour category. I am trying to find related results when balls can belong to multiple categories. $\endgroup$ – Ben Feb 25 at 10:07
  • $\begingroup$ The multivariate hypergeometric distribution allows us to compute the joint probability mass function for the $2^p$ category combinations but not, without some brute-force summation, the marginal probability mass function for the $p$ colour counts. $\endgroup$ – Ben Feb 25 at 10:18
  • $\begingroup$ I see. Sorry I misunderstood your question. This is not an easy problem and at this point I don't have much to contribute. $\endgroup$ – Lee David Chung Lin Feb 25 at 11:57

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