Compute the expected number of tests necessary for each group.?

Question

To determine whether of not they have a certain disease, 100 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 analyzed together. If the test is negative, one test will be suffice for the 10 people: 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 disease is 0.1 for all people, compute the expected number of tests necessary for each group.

Answer 1

For each group of ten:

the event 'no-one has the disease' occurs with probability 0.9^10 and involves only one test;

the event 'at least one person has the disease' occurs with probability 1 - 0.9^10 and involves eleven tests.

Thus, the expected number of tests for one group is

1*0.9^10 + 11*(1 - 0.9^10)

= 7.51 (2 d.p.).

If you want the expected number for all ten groups, then it's 75.13 (2 d.p.).

Answer 2

P (all negative in one group) = 0.9 ^ 10 = 0.3486784401

P (not all negative in one group) = 1 - 0.3486784401 = 0.6513215599

1 test if all negative, 11 tests if not all negative, so

expected number of tests = 0.3486784401 x 1 + 0.6513215599 x 11

= 0.3486784401 + 7.1645371589

= 7.51321559