Blood is taken from each of n individuals to be tested for a certain disease. Rather than test each sample separately, a pooled method is used in an attempt to reduce the number of tests needed. Part of each blood sample is taken, and these parts are combined to form a pooled sample. The pooled sample is then tested. If the result is negative, then none of the n individuals has the disease, and no further tests are needed. If the pooled sample tests positive, then each individual is tested to see which of them have the disease.
a. Let X represent the number of tests that are carried out. What are the possible values of X?
b. Assume that n = 4 individuals are to be tested, and the probability that each has the disease, independent of the others, is p = 0.1. Find μX.
c. Repeat part (b) with n = 6 and p = 0.2.
d. Express μX as a function of n and p.
e. The pooled method is more economical than performing individual tests if μX < n. Suppose n = 10. For what values of p is the pooled method more economical than performing n individual tests?