To determine whether or not they have a certain desease, 260 people are to have

Question

To determine whether or not they have a certain desease, 260 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 13. The blood samples of the 13 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 13 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 13 people will also be individually tested and, in all, 14 tests will be made on this group. Assume the probability that a person has the desease is 0.07 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 260 people.

Explanation / Answer

Given that,

number of people are to have their blood tested = 260

Group size = 13

P(person has disease) = 0.07

P(person has no disease) = 1-0.07 = 0.93

For each group of 13:

the event 'no-one has the disease' occurs with probability 0.93^13 and involves only one test;

the event 'at least one person has the disease' occurs with probability 1 - 0.93^13 and involves 14 tests.

Thus, the expected number of tests for one group is

1*0.93^13 + 14*(1 - 0.93^13) = 0.3893 + 14*0.6107 = 0.3893 + 8.5499

expected number of tests for one group is 8.9392

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