I'm trying to find out the name of a distribution that is like negative binomial, only for finite population and without replacement. Or like Hypergeometric distribution where the last event has to be a success.

That is:

Let's say we have $N$ balls in an urn, where $W$ of them are white balls and $B$ are black balls. I want to know what are the chances of needing to draw exactly $n$ balls so it is the first time I got $k$ of the $n$ drawn balls to be black (e.g: that the last ball drawn was black, and a total of $k$ out of $n$ are black).

I can't seem to find the name of such a distribution, so if you could also advise me on where to find names of distributions, that would also be lovely.



The usual name for the distribution is Negative Hypergeometric Dstribution.

We want the probability that exactly $k-1$ of the first $n-1$ are black and the $n$-th is black.

The probability of exactly $k-1$ black when $n-1$ balls are drawn can be written down using standard formulas, for it is a straight hypergeometric problem.

Given that this has happened, we know the number of balls remaining, and the number of black balls remaining, so we know the (conditional) probability the $n$-th draw is black. Multiply.

  • $\begingroup$ Bingo. Many thanks Andre. $\endgroup$ – Tal Galili Apr 9 '13 at 5:53

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