Question

In the lecture, we covered the “pooled-testing” problem, namely when you do blood test on large number of people, it is more efficient to pool k people’s blood together to do the test: if this pooled blood sample results in negative, then you know all these k people are negative, if this pooled blood sample results in positive, then you need to re-test each one of them in this group. Therefore, for each group of k people, you either test once or k+1 times. Suppose p is the rate of certain disease, in case p= 0.1, in the class, we did the calculation 0f E(X), where x is the random number of times you need to do the test. For this problem, if P= 0.05 and N=150000, do the calculation of E(X)  for k =3, 4, 5, 6, 7, 8, 10, 15, 25, 35, and then determine the best k.

Answer 1

is the rate of certain disease and

The probability that selected people do not have the disease is . So we have to conduct

1 test with probability and

tests with probability .

The number of groups in the test is

So the expected number of test is

The expeceted values for is computed below.

k = 3,  71393.75 
k = 4,  65324.06 
k = 5,  63932.86  
k = 6,  64736.22 
k = 7,  66677.98 
k = 8,  69236.94  
k = 10, 75189.46 
k = 15, 90506.32 
k = 25, 114391.56 
k = 35, 129373.21

The expeceted number of tests is minimum when which the best .