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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)?
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