Group Testing, the Pooled Hypergeometric Distribution, and Estimating the Number of Defectives in Small Populations

The testing of combined bacteriological samples - or "group testing" - was introduced to reduce the cost of identifying defective individuals in populations containing small proportions of defectives. It may also be applied to plants, animals, or food samples to estimate proportions infected, or to accept or reject populations. Given the proportion defective in the population, the number of positive combined samples is approximately binomial when the population is large: we find the exact distribution when groups include the same number of samples. We derive some properties of this distribution, and consider maximum-likelihood and Bayesian estimation of the number defective.