(Feller14) A large number, N, of people are subjected to a blood test. This can be administered in two ways: (1) Each person can be tested separately, in this case N test are required, (2) the blood samples of k persons can be pooled and analyzed together. If this 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 that the probability p that a test is positive is the same for all people and that these events are independent.
(a) Find 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 (2)? (Assume that N is divisible by k.)
(c) For small p, show that the value of k which will minimize the expected number of tests under the second plan is approximately 1/pp.