# Solved – Sum or mean of several related hypergeometric distributions

I have an odd problem which can be phrased in a general way, and a more specific way. I'm curious about the answers to both. Although, really, it's the k=0 case that I'm really interested in – deriving the answer with respect to some properties of the distribution of m.

These may be totally basic, so, apologies if they are.

1) I have several different hypergeometric distributions, H(k; N, m, n) where k is the number of 'success' draws, N is the population size, m is the number of possible success draws, and n is the total number of draws. Each distribution has a different value for m, but all else is the same. Is there an easy way to either sum them up or provide a more compact notation for them?

Or, to phrase it as a ball and urn question, I have many boxes, each with N balls. In each urn i, m(i) balls are white and the rest are black. If I take n draws from each urn in turn, what is the average probability of k white balls drawn in each urn? (indeed, is there a distribution for this – there must be)

2) More specifically, I'm interested in the case where k=0. This actually reduced quite nicely to

\$binom{N-m}{n}/binom{N}{n}\$

But…again, I want to sum over a lot of different values of m from different members of a population. Is there a way to get at this with, say, an average value of m or otherwise?

Again, in box urn terms, this would be the average probability of drawing NO white balls from any of the urns.

p.s. I have actually always wondered a similar thing for the binomial distribution and suspect the answers may be related. Sure, B(a,p)+B(b,p) = B(a+b,p) where a and b are the number of trials and p is the probability of success. But what about B(a,p1)+B(a,p2)?

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