Suppose that a large group of people are undergoing a blood test for a particular illness. The probability that a random person has the illness in question is equal to 0.001. In order to save on the work, it is decided to split the group into smaller groups each consisting of r people. The blood samples of the r people are then mixed and tested all at once. If the test results are favorable, then one test will have been sufficient for that whole group. Otherwise, r extra tests will be necessary in order to test each of the r people individually. What is the expected value of the number of tests that will have to be done for a group of r people? Verify that r = 32 is the optimal group size.