  We consider the optimal strategy for laboratory testing of biological samples
when we wish to know the results for each sample rather than the average
prevalence of positive samples. If the proportion of positive samples is low
considerable resources may be devoted to testing samples most of which are
negative. An attractive strategy is to pool samples. If the pooled samples test
positive one must then test the individual samples, otherwise they can all be
assumed to be negative. The pool should be big enough to reduce the number of
tests but not so big that the pooled samples are almost all positive. We show
that if the prevalence of positive samples is greater than 30% it is never
worth pooling. From 30% down to 1% pools of size 4 are close to optimal. Below
1% substantial gains can be made by pooling, especially if the samples are
pooled twice. However, with large pools the sensitivity of the test will fall
correspondingly and this must be taken into consideration. We derive simple
expressions for the optimal pool size and for the corresponding proportion of
samples tested.
