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

total we have 260 people

each group contain 13 people therefore 20 groups are there in total

also if test negative total test = 1

if test positive total test = 14

probability of having disease = 0.07

probability of not having disease = 0.93

now

probability that 1 groups shows positive test = 0.07 ( because even 1 disease person is enough)

probability that 1 groups shows negative test = 0.93^13 = 0.389 ( here all should be sisease free)

expected number of test = 14*20*0.07 + 1*20*0.389 = 27.38

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