Hypergeometric distribution of the number of draws from an urn with two types of items before one of the counts reaches a threshold

Authors: J. L. GONZALEZ-SANTANDER

Abstract: We consider an urn with $R$ elements of one type and $B$ elements of other type. We calculate the probability distribution $P_{n_{R},n_{B}}^{R,B}\left( s\right) $ wherein the random variable $s$ is the number of draws from the urn until we reach $n_{R}$ elements of type $R$ or $n_{B}$ elements of type $B$. We calculate the mean value $\left\langle s\right\rangle $ and the standard deviation $\sigma $ of $P_{n_{R},n_{B}}^{R,B}\left( s\right) $ in terms of hypergeometric functions. For $n_{R}=n_{B}$ and $B=R$ , we reduce $\left\langle s\right\rangle $ and $\sigma $ in terms of elementary functions. Also, the normalization condition leads to a new hypergeometric summation formula involving $_{3}F_{2}$ terminating series with unity argument. For $n_{R}=n_{B}$, we provide an alternative proof of this summation formula using $q$-hypergeometric functions. As a consistency test, computer simulations have been performed to confirm the analytical results obtained.

Keywords: Generalized hypergeometric function, hypergeometric probability distribution, $q$-hypergeometric function

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