Question

To determine whether 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 place the peopleinto groups of 10. The blood samples of the 10 people in each group will be pooled and analyzedtogether. If the test is negative, one test will 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 that the probability that a person has the disease is .1 for all people, independently of eachother, and compute the expected number of tests necessary for each group. (Note that we areassuming that the pooled test will be positive if at least one person in the pool has the disease.)

Answer 1

Answer:

The expected number of tests necessary for each group is 7.51.

Step-by-step explanation:

The probability of a person having the disease is, P (X) = 0.10.

Then the probability of a person not having the disease is P (X') = 0.90.

It is provided that if the test is negative, one test will suffice for the 10 people.

Bur 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.

Compute the expected number of tests necessary for each group as follows:

Expected number of tests = Negative Result × P (X') + Positive Result × P (X)

                                           $=(1\times (0.90)^{10})+(11\times [1-(0.90)^{10}])\\=0.3486784401+7.1645371589\\=7.513215599\\\approx 7.51$

Thus, the expected number of tests necessary for each group is 7.51.