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Specimens are collected from $N$ different sources. Each specimen hasprobability $p$ of being contaminated (e.g., in the case of an infectiousdisease, $p$ is the prevalence rate), independently of the other specimens. Inmany cases group testing is applicable, namely one can take small portions fromseveral specimens, mix them together, and test the mixture for contamination,so that if the test turns positive, then at least one of the samples in themixture is contaminated. In this paper we give a detailed probabilisticanalysis of a binary search scheme, we propose, for determining allcontaminated specimens. More precisely, we study the number $T(N)$ of testsrequired in order to find all the contaminated specimens, if this search schemeis applied. We derive recursive and, in some cases, explicit formulas for theexpectation, the variance, and the characteristic function of $T(N)$. Also, wedetermine the asymptotic behavior of the moments of $T(N)$ as $N \to \infty$and from that we obtain the limiting distribution of $T(N)$ (appropriatelynormalized), which turns out to be normal.

A binary search scheme for determining all contaminated specimens