Question

A number, \(k\), of people are subjected to a blood test, the result of which is either "positive" or "negative". It can be processed in two ways:

i). Each person can be tested separately, so \(k\) tests are required.
ii). The blood samples of all \(k\) people can be pooled and analyzed together. If this test is negative, then one test suffices for the \(k\) people, while if the test is positive, each of the \(k\) people must be tested separately so \(k+1\) tests are then required.
Assume that the probability, \(p\), that a test is positive is the same for all people and that these events are all independent.
a). Find the probability that the test for a pooled sample of \(k\) people will be positive.
b). What is the expected value of the number of tests necessary under plan ii)?