A binary search scheme for determining all contaminated specimens

TL;DR

This paper analyzes a probabilistic binary search method for identifying all contaminated specimens among many, providing formulas for the expected number of tests, variance, and the distribution of tests needed.

Contribution

It introduces a detailed probabilistic analysis of a binary search scheme for detecting all contaminated specimens, including explicit formulas and asymptotic behavior.

Findings

Expected number of tests derived

Variance and distribution of test count characterized

Asymptotic normality established

Abstract

Specimens are collected from $N$ different sources. Each specimen has probability $p$ of being contaminated (e.g., in the case of an infectious disease, $p$ is the prevalence rate), independently of the other specimens. In many cases group testing is applicable, namely one can take small portions from several 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 the mixture is contaminated. In this paper we give a detailed probabilistic analysis of a binary search scheme, we propose, for determining all contaminated specimens. More precisely, we study the number $T(N)$ of tests required in order to find all the contaminated specimens, if this search scheme is applied. We derive recursive and, in some cases, explicit formulas for the expectation, the variance, and the characteristic function of $T(N)$. Also, we determine the asymptotic behavior of the moments of $T(N)$ as $N \to \infty$ and from that we obtain the limiting distribution of $T(N)$ (appropriately normalized), which turns out to be normal.