Question

To determine whether or not they have a certain desease, 304 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 19. The blood samples of the 19 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 19 people (we are assuming that the pooled test will be positive if and only if at least one person in the pool has the desease); whereas, if the test is positive each of the 19 people will also be individually tested and, in all, 20 tests will be made on this group. Assume the probability that a person has the desease is 0.04 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 304 people.

Answer 1

The expected number of tests necessary for the entire group of 304 people is 180.0325

Explanation:

Probability of person being positive = p = 0.04

Probability of person being negative = q = 1-0.04=0.96

Number of people in each group = 19

We will use binomial over here

Probability of no test positive :

`P(x=0) = ^(19)C_0 (0.04)^0 (0.96)^(19) =0.460419201958`

P( at least one tests positive) = 1-P( no one tests positive)= 1-0.460419201958= 0.539580798042

Expected number of tests for each group = 1(0.460419201958)+20(0.539580798042) = 11.2520351628

Number of groups =
`\frac{\text{Total population}}{\text{no. of people in each group}}=(304)/(19)=16`

Expected number of tests necessary for the entire group of 304 people:

=
`\text{Expected number of tests for each group} * \text{No. of groups}`

=
`11.2520351628 * 16`

=180.0325

Hence The expected number of tests necessary for the entire group of 304 people is 180.0325