Consider a disease whose presence can be identified by carrying out a blood test. Let p denote the probability that a randomly selected individual has the disease. Suppose n individuals are independently selected for testing. One way to proceed is to carry out a separate test on each of the n sample. A potentially more economical approach, group testing, was introduced during World War II to identify syphilitic men among army inductees. First, take a part of each blood sample, combine these specimens, and carry out a single 1 test. If no one has the disease, the result will be negative, and only the one test is required. If at least one individual is sick, the test on the combined sample will yield a positive result, in which case the n individual tests are the carried out. If p = .1:
What is the expected number of tests using this procedure when n = 3? Is this procedure better on average than simply testing everyone?