The blood-testing problem.12 A large number, N, of people are subject to a blood test. This can be administered in two ways. (i) Each person can be tested separately. In this case N tests are required. (ii) The blood samples of k people can be pooled and analyzed together. If the test is negative, this one test suffices for the k people. If the test is positive, each of the k persons must be tested separately, and in all k +1 tests are required for the k people. Assume the probability p that the test is positive is the same for all people and that people are stochastically independent.

(a) What is the probability that the test for a pooled sample of k people will be positive?

(b) What is the expected value of the number, X, of tests necessary under plan (ii)?

(c) Find an equation for the value of k which will minimize the expected number of tests under the second plan.

(d) Show that this k is close to 1/p, and hence that the minimum expected number of tests is about 2Np. (This remark is due to M. S. Raff.)