It is known that 5% of a population have disease A, which can be detected by a blood test. Suppose that N (a large number) people are to be tested. This can be done in two ways: Either (1) each person is tested separately or (2) the blood samples of k people are pooled together and analyzed. (Assume that N = nk, with n an integer.) In method 2, if the test is negative, all the people in the pool are healthy (that is, just this one test is needed). If the test is positive, each of the k persons must be tested separately (that is, a total of k + 1 tests is needed).
a For fixed k, what is the expected number of tests needed in method 2?
b Find the k that will minimize the expected number of tests in method 2.
c How many tests does part (b) save in comparison with part (a)?