When screening a large population for a rare disease, scientists often use a scheme called pooled sampling to get faster and cheaper results. They pool samples from (say) 100 people at a time, and test for the disease in the pooled sample, which is called a batch. If the batch tests negative for the disease, then all 100 of the contributors must be negative. If the batch tests positive, then all 100 individual samples within the batch are tested individually to identify which of them are positive.

Suppose investigators need to test blood samples from 20,000 people for a disease with an overall prevalence of 0.4%. You can assume that each individual blood sample is positive with probability 0.004, and that all samples are independent.

a. Using 100 samples per batch, let X be the number of positive samples in one batch. What is the distribution of X?

b. What is the probability that a single batch of 100 samples has no positive contributors? Write your answer in terms of a probability involving X, and calculate the answer.

c. What is the probability that a single batch tests positive?
Again, write your answer in terms of X, and calculate it.

d. Let N be the number of positive batches out of the total of 200 batches. What is the distribution of N?

e. What is the expected number of positive batches out of 200? Write your answer in terms of N, and calculate it.

f. Find an expression involving X and/or N for the expected number of tests that the investigators have to perform, to fully test all 20,000 people. Calculate the answer. How does it compare with the 20,000 tests that would need to be done if the investigators didn’t use the pooled sampling scheme?
g. We now want to know if the answer from part

(f) is the best we can do. Rather than 100 samples per batch, suppose we use b samples per batch. Follow your working above to show that the expected number of tests needed is E(number of tests for batch size b)
= 20000 
1 +
1 b − 0.996b 
.
Check that this formula gives your answer from part (f)
when b = 100.