In an army camp, a large number, N, of soldiers are subject to a blood test to examine whether they have a certain disease. There are two ways to carry out this experiment:

A. each soldier is tested separately, so that N tests take place;

B. the blood samples of k soldiers (k < N) can be pooled and analyzed together.

If the test outcome is negative, this means that none of the soldiers in this group have the disease. If it is positive, each of the k persons is then sampled individually, so that k + 1 tests are required for a group of k people.3 Assume that each soldier has a probability p of having the disease and that test outcomes for different soldiers are independent.

(i) What is the probability that the test for a pooled sample of k soldiers will be positive?

(ii) Find the expected value of the number, X, of blood tests required under Plan B above.