Question

To determine whether or not they have a certain disease, 160 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 suffice for the 10 people (we are assuming that the pooled test will be positive if and only if at least one person in the pool has the disease); 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 0.06 for all people, independently of each other.

Compute the expected number of tests necessary for each group.

Answer 1

Answer:

Given,

Probability of positive result = 0.06

Probability of negative result = 1 - 0.06

= 0.94

Consider,

P(x = 1) = P(Just one test required for the gathering)

= P(Every one of 10 individuals in the gathering shows negative outcome)

= 0.94*0.94*........*0.94 [ 10 times]

= 0.94^10

= 0.5386

P(x = 11) = P(At least one shows +ve)

= 1 - P(no one shows +ve)

= 1 - (0.94^10)

= 1 - 0.5386

= 0.4614

Expected number of tests necessary for each group = x1*P(x1) + x2*P(x2)

= 1*0.5386 + 11*0.4614

= 5.614

Expected number of tests necessary for each group = 6