In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without replacement in which each sample can be classified into two mutually exclusive categories like Pass/Fail or Employed/Unemployed. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw. Unlike the standard hypergeometric distribution, which describes the number of successes in a fixed sample size, in the negative hypergeometric distribution, samples are drawn until r {\displaystyle r} failures have been found, and the distribution describes the probability of finding k {\displaystyle k} successes in such a sample. In other words, the negative hypergeometric distribution describes the likelihood of k {\displaystyle k} successes in a sample with exactly r {\displaystyle r} failures.
Definition There are N {\displaystyle N} elements, of which K {\displaystyle K} are defined as "successes" and the rest are "failures". Elements are drawn one after the other, without replacements, until r {\displaystyle r} failures are encountered. Then, the drawing stops and the number k {\displaystyle k} of successes is counted. The negative hypergeometric distribution, N H G N , K , r ( k ) {\displaystyle NHG_{N,K,r}(k)} is the discrete distribution of this k {\displaystyle k} .
The negative hypergeometric distribution is a special case of the beta-binomial distribution with parameters α = r {\displaystyle \alpha =r} and β = N − K − r + 1 {\displaystyle \beta =N-K-r+1} both being integers (and n = K {\displaystyle n=K} ). The outcome requires that we observe k {\displaystyle k} successes in ( k + r − 1 ) {\displaystyle (k+r-1)} draws and the ( k + r ) -th {\displaystyle (k+r){\text{-th}}} bit must be a failure. The probability of the former can be found by the direct application of the hypergeometric distribution ( H G N , K , k + r − 1 ( k ) ) {\displaystyle (HG_{N,K,k+r-1}(k))} and the probability of the latter is simply the number of failures remaining ( = N − K − ( r − 1 ) {\displaystyle =N-K-(r-1)} ) divided by the size of the remaining population ( = N − ( k + r − 1 ) {\displaystyle =N-(k+r-1)} ). The probability of having exactly k {\displaystyle k} successes up to the r -th {\displaystyle r{\text{-th}}} failure (i.e. the drawing stops as soon as the sample includes the predefined number of r {\displaystyle r} failures) is then the product of these two probabilities:
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