Suppose that a blood test for a disease must be given to a population of N peopl

Question

Suppose that a blood test for a disease must be given to a population of N people, where N is large. At most N individual blood tests must be done. The following strategy reduces the number of tests. If 100 people are selected from the population and their blood samples are pooled, one test will determine whether any of the 100 people test positive. If the test is positive, those 100 people are tested individually, making 101 tests necessary. However, if the pooled sample tests negative, then 100 people have been conclusively tested with only one test. Probability theory shows that if the pool group size is x then the average number of blood tests required to test N people is N(1 - q^x + 1/x), where q is the probability that any one person tests negative. What group size x minimizes the average number of tests in the case where N = 10,000 and q = 0.95?

Explanation / Answer

Given

probability P =N(1-q^x + 1/x)

=> P = 10000(1-0.95^x +1/x)

by trial and error taking x=1 to 9

atx= 1 P = 10500

at x=5 P = 4260

at x= 6 P = 4315

at x=9 P =4796

so the miminal value occurs when the group size is 5

=> x=5

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