Consider the situation where we are sampling without replacement from a population *M* with *D* items with the characteristic of interest until we have *s* items with the required characteristic. The distribution of the number of trials we will need to get *s* success can be easily calculated in the same manner as we developed the Negative Binomial distribution. The probability of observing (*s*-1) successes in (*x*-1) trials (i.e. we have had (x-1) - (s-1) = x -s failures) is given by direct application of the Hypergeometric distribution:

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p(x,s-1)=\frac{\left( \begin{array}{c} D \\ s-1 \end{array} \right) \left( \begin{array}{c} M-D \\ x-s \end{array} \right)}{\left( \begin{array}{c} M \\ x-1 \end{array} \right) } |

The probability *p* of then observing a success in the next trial (the *x*th trial), is simply the number of *D* items remaining (=*D*-(*s*-1)= D-s+1) divided by the size of the population remaining (= *M*-(*x*-1)=M-x+1):

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p=\frac{\big(D-s+1 \big)}{\big(M-x+1\big)} |

and the probability of needing exactly *x* trials to obtain *s* success, where trials are stopped at the *s*th success, is then the product of these two probabilities:

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p(x,s)=\frac{\left( \begin{array}{c} D \\ s-1 \end{array} \right) \left( \begin{array}{c} M-D \\ x-s \end{array} \right) (D-s+1)}{\left( \begin{array}{c} M \\ x-1 \end{array} \right)(M-x+1) } |

This is the probability mass function for the Inverse Hypergeometric distribution InvHyperGeo(*s,D,M*) and is analogous to the Negative Binomial distribution for the binomial process and the Gamma distribution for the Poisson process. So:

*n* = InvHyperGeo(*s*,*D*,*M*)

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For a population *M* that is large compared to *s*, the Inverse Hypergeometric distribution is closely approximated by the Negative Binomial:

InvHypergeo(*s,D,M*) *»* NegBinomial(*D*/*M,s*)

and if the probability *D*/*M* is very small:

InvHypergeo(*s,D,M*) *»* Gamma(s,*M*/*D,s*)

The four figures below show examples of the Inverse Hypergeometric distribution. In the first figure you can see the probability mass function of the number of trials needed for getting 4 successes when drawing samples from a population 50 in which 5 individuals have the characteristic you are interested in. We leave to you the task to explain in words the figures 2 – 4.

An Inverse Hypergeometric distribution is sometimes called a *Negative Hypergeometric* distribution.

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