To determine whether or not they have a certain desease, 70 people are to have t

Question

To determine whether or not they have a certain desease, 70 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group the people in groups of 10. The blood samples of the 10 people in each group will be pooled and analized together. If the test is negative. one test will suffice for the 10 people (we are assuming that the pooled test will be positive if and only if at least one person in the pool has the desease); whereas, if the test is positive each of the 10 people will also be individually tested and, in all, 11 tests will be made on this group. Assume the probability that a person has the desease is 0.08 for all people, independently of each other, and compute the expected number of tests necessary for the whole group of 70 people.

Explanation / Answer

For a group of 10 people the probability that at least one person has the disease equals 1 - the probability that none of the 10 have the disease. For each person, the probability of not having the disease is (1 - 0.03) = 0.97. Because whether a person has the disease is independent of everyone else, the probability of no one in the group having the condition is 0.97 ^ 10 = 0.73742412689492826049. The expected number of tests is 1 * Pdf + 11 (1 - Pdf) = 11 - 10 * Pdf = 3.6257587310507173951.

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