Suppose the blood of 1000 persons has to be tested to see which ones are infected by a (rare) disease. Suppose that the probability that the test is positive is p = XXXXXXXXXXThe obvious way to...

Suppose the blood of 1000 persons has to be tested to see which ones are infected by a (rare) disease. Suppose that the probability that the test is positive is p = 0.001. The obvious way to proceed is to test each person, which results in a total of 1000 tests. An alternative procedure is the following. Distribute the blood of the 1000 persons over 25 groups of size 40, and mix half of the blood of each of the 40 persons with that of the others in each group. Now test the aggregated blood sample of each group: when the test is negative no one in that group has the disease; when the test is positive, at least one person in the group has the disease, and one will test the other half of the blood of all 40 persons of that group separately. In total, that gives 41 tests for that group. Let Xi be the total number of tests one has to perform for the ith group using this alternative procedure.

a. Describe the probability distribution of Xi, i.e., list the possible values it takes on and the corresponding probabilities.

b. What is the expected number of tests for the ith group? What is the expected total number of tests? What do you think of this alternative procedure for blood testing?