What is an Hypergeometric distribution where the last event must be a success?

Question

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.

Thanks.

Answer 1

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.