(Feller) A large number, [math]N[/math], 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 [math]N[/math] test are required, (2) the blood samples of [math]k[/math] persons can be pooled and analyzed together. If this test is negative, this one test suffices for the [math]k[/math] people. If the test is positive, each of the [math]k[/math] persons must be tested separately, and in all, [math]k + 1[/math] tests are required for the [math]k[/math] people. Assume that the probability [math]p[/math] that a test is positive is the same for all people and that these events are independent.

• Find the probability that the test for a pooled sample of [math]k[/math] people will be positive.
• What is the expected value of the number [math]X[/math] of tests necessary under plan (2)? (Assume that [math]N[/math] is divisible by [math]k[/math].)
• For small [math]p[/math], show that the value of [math]k[/math] which will minimize the expected number of tests under the second plan is approximately [math]1/\sqrt p[/math].

Notes