A large number N of people are subjected to a blood investigation to test for the presence of an illegal drug. This investigation is carried out by mixing the blood of k persons at a time and testing the mixture. If the result of the analysis is negative then this is sufficient for all k persons. If the result is positive then the blood of each person must be analysed separately, making k + 1 analyses in all. Assume that the probability p of a positive result is the same for each person and that the results of the analyses are independent. Find the expected number of analyses, and minimize with respect to k. In particular, find the optimum value of k when p = 0.01, and the expected saving compared with a separate analysis for all N people.