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The probability to have $k$ white balls in a sample of size $n$ taken from an urn of $N$ balls with $K$ of them being white is equal to: $$ P(k|n,N,K) = \frac{{{n}\choose{k}}{{N-n}\choose{K-k}}}{{{N}\choose{n}}} $$ How to infer a probability when $K$ is not determined a priori.

In other words the question would be: What is the probability to have $k$ white balls in a sample of size $n$ taken from an urn of $N$ balls with an unknown amount of them being white (I only know for sure that $k$ are definitely white but it can be more).

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You can estimate $K$ using method of moments estimator

$$ \frac{k}{n} \approx \frac{K}{N} \implies \frac{N}{n} k $$

or maximum likelihood estimator as described by Zhang (2009):

$$ \frac{N-1}{n} k $$

for derivation and further details check the following paper:

Zhang, H. (2009). A note about maximum likelihood estimator in hypergeometric distribution. Comunicaciones en Estadística, 2(2), 169-174.

On another hand, it you want to define a distribution of $k$ white balls, drawn without replacement from the urn containing $N$ balls in total, while treating the total number of white balls $K$ as unknown, i.e. as a random variable, then you can define such problem in terms of Bayesian model, with beta-binomial prior (in fact a conjugate prior) for $K$ (as described by Fink, 1997 and Dyer and Pierce, 1993):

$$ k \sim \mathcal{H}(N,K,n) \\ K \sim \mathcal{BB}(N, \alpha, \beta) $$

what follows to a beta-binomial posterior predictive distribution of $k$ parametrized by $N$, $\alpha' = \alpha + k$ and $\beta' = \beta + N-k$, and the posterior distribution of $K$ is

$$ f(K\mid k,N,\alpha,\beta) = {N-n \choose K-k} \frac{\Gamma(\alpha+K)\,\Gamma(\beta+N-k)\,\Gamma(\alpha+\beta+n)}{\Gamma(\alpha+k)\,\Gamma(\beta+n-k)\,\Gamma(\alpha+\beta+N)} $$

If you want to assume that $K$ can be anything in the $[k, N-n+k]$ range, you can use uniform $\alpha=\beta=1$ prior. For further details check:

Dyer, D. and Pierce, R.L. (1993). On the Choice of the Prior Distribution in Hypergeometric Sampling. Communications in Statistics - Theory and Methods, 22(8), 2125-2146.

You may also be interested in reading about capture-recapture method where you are interested in finding $N$, since it is closely related and follows the same logic.

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  • $\begingroup$ How did you compute here the predictive posterior of $k$ (the beta-binomial parametrized by $N$, $\alpha'$, and $\beta'$)? I couldn't find any info about this in the references. $\endgroup$ – pms Aug 12 at 2:50
  • $\begingroup$ @pms sorry, forgot to add the link, now updated $\endgroup$ – Tim Aug 12 at 10:39
  • $\begingroup$ Actually, I found and checked previously Fink's compendium, but I couldn't find there anything about the predictive posterior. In this case, the predictive posterior is a beta-binomial mixture of hypergeometric distributions, but you wrote here that it's the beta-bionomial with parameters $N$, $\alpha'$, and $\beta'$, so I'm wondering how did you figure this out; is this conclusion straightforward? $\endgroup$ – pms Aug 13 at 21:44
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    $\begingroup$ @pms have you checked the paper by Dyer and Pierce? $\endgroup$ – Tim Aug 14 at 5:58
  • $\begingroup$ Indeed, in Dyer and Pierce there is an expression for marginal distribution of $k$ (their $m(x)$ at the top of page 2131), but it's not explained how it was obtained. I imagine it can be calculated by marginalizing hypergeometric likelihood over the beta-binomial prior, but this calculation looks a bit tedious at first sight. Btw. A small correction -- it should be $\beta'=\beta+n-k$ in your above answer. Everything else looks fine. $\endgroup$ – pms Aug 16 at 21:50

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