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

Abstract

We consider an urn with R elements of one type and B elements of other type. We calculate the probability distribution P R,B nR,nB (s) wherein the random variable s is the number of draws from the urn until we reach nR elements of type R or nB elements of type B . We calculate the mean value ⟨s⟩ and the standard deviation σ of P R,B nR,nB (s) in terms of hypergeometric functions. For nR = nB and B = R, we reduce ⟨s⟩ and σ in terms of elementary functions. Also, the normalization condition leads to a new hypergeometric summation formula involving 3F2 terminating series with unity argument. For nR = nB , 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.