Chapter 4.5, Problem 44E

Blood testing Suppose a blood test for a disease is given to a population of N people, where N is large. At most, N individual blood tests must be done. The following strategy reduces the number of tests. Suppose 100 people are selected from the population and their blood samples are pooled. One test determines whether any of the 100 people test positive. If that test is positive, those 100 people are tested individually, making 101 tests necessary. However, if the pooled sample tests negative, then 100 people have been tested with one test. This procedure is then repeated. Probability theory shows that if the group size is x (for example, x = 100, as described here), then the average number of blood tests required to test N people is N(1 − q^x + 1/x), where q is the probability that any one person tests negative. What group size x minimizes the average number of tests in the case that N = 10, 000 and q = 0.95? Assume x is a real number between 1 and 10,000.