A medical laboratory tests many blood samples for a certain disease that occurs in about 5% of the samples. The lab collects samples from 10 persons and mixes together some blood from each sample. If a test on the mixture is positive, an additional 10 tests must be run, one on each individual sample. But if the test on the mixture is negative, no other tests are needed. It can be shown that the test of the mixture will be negative with probability (.95)10 = .599, because each of the 10 samples has a 95% chance of being free of the disease. If X is the total number of tests required, X has the probability table shown in Table 1.

(a) Find E(X).

(b) If the laboratory uses the procedure described on 200 blood samples (that is, 20 batches of 10 samples), about how many tests can it expect to run?

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Table 1 Blood Test Probabilities for Batches of 10 Samples Total tests Probability Test of Mixture Negative 1 .599 Positive 11 .401