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To determine whether or not they have a certain disease, 140 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 0.07 for all people, independently of each other.
Compute the expected number of tests necessary for each group.
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Consider one group of 10 people.
The probability that one grouped test will show that no one person of 10 persons in the group
has disease is
P(no disease in the group of 10) = = = 0.484 (rounded).
The probability that at least one in the group of 10 has the disease is the complement of it,
or 1 - 0.484 = 0.516.
Thus, 1 comprehensive test for the group of 10 is enough with the probability 0.484,
and 10 additional tests are needed with the probability of 0.516.
The expected number of tests is then 1*0.484 + 10*0.516 = 5.644 for each such group of 10 persons. ANSWER
Solved.