To determine whether or not they have a certain desease, 210 people are to have
ID: 3336182 • Letter: T
Question
To determine whether or not they have a certain desease, 210 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 14. The blood samples of the 14 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 14 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 14 people will also be individually tested and, in all, 15 tests will be made on this group. Assume the probability that a person has the desease is 0.06 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 210 people.
Explanation / Answer
To determine whether or not they have a certain desease, 210 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 14. The blood samples of the 14 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 14 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 14 people will also be individually tested and, in all, 15 tests will be made on this group. Assume the probability that a person has the desease is 0.06 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 210 people.
Let X be the number of tests needed for each group of 14 people. Then, if nobody has the disease 1 test is enough. But if the test is positive then there will be 15 test (1 + 14).
Probability of disease=0.06
Probability of no disease=0.94
The probability distribution of X is:
x
P(x)
1
0.94^14
15
1-0.94^14
Expected number of tests for 14 persons = 1*(0.94^14) +14*(1-0.94^14)
=8.533199
=8.53
For the entire group of 210, expected number of tests = (210/14)*8.53
=127.95
x
P(x)
1
0.94^14
15
1-0.94^14
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