Question 1188512: To determine whether or not they have a certain desease, 204 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 17. The blood samples of the 17 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 17 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 17 people will also be individually tested and, in all, 18 tests will be made on this group. Assume the probability that a person has the desease is 0.1 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 204 people.
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Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Probability nobody will test positive in a group of 17 is 0.9^17=0.1668
Probability at least one will test positive is the complement or 0.8332
Expected value for 12 groups is 0.1668*12=2.00 + 0.8332(12)=10.00
so expect 2 groups to have no positive tests and therefore 34 people have required 2 samples to be run.
The other 170 fit in the 10 tests where at least one was positive and they all had to be run separately. This is 10 tests + 170 separate=180 tests
The total expected number of tests is 182.
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