Investigation 8: Composite Blood Testing (due Tuesday, October 12)

You may work with one other person on this assignment, handing in one report with both names. Word-processed reports are preferred to hand-written ones. Integrate appropriate computer output into your report.

Suppose that in a group of 10 people, each one has a 0.1 probability of having a certain disease, independently from person to person. In order to identify who has the disease, we could proceed by conducting a blood test on each individual. An alternative approach, introduced during World War II to identify syphilitic men among army inductees, is to pool blood samples from all ten people and run a test on that composite blood sample. This composite test gives a positive result if at least one person has the disease, and it gives a negative result if nobody has the disease. If this composite test is positive, then each person would need to have an individual blood test conducted.

a) Let X be the total number of tests required with the “composite testing” plan. Determine the probability distribution of X. [Hint: There are only two possible value of X. Figure out what they are, and then determine the probability of each.]

b) Determine the expected value of the number of tests required with the “composite testing” plan. Is this less than the number of tests that are required if each individual is tested separately? Explain.

c) Does the composite testing plan guarantee that fewer tests will be required than with the alternative of testing each individual in the first place? If not, what is the probability that the composite testing plan will reduce the number of tests required?

Now suppose that the probability of having the disease is given by p for each person, again independently from person to person.

d) Express the expected number of tests under the composite plan as a function of p.

e) Determine the values of p for which the expected number of tests with the composite plan is less than the ten tests required with individual testing.

Finally, suppose that the group consists of n people and that each has probability p of having the disease, independently from person to person.

f) Express the expected number of tests under the composite plan as a function of both n and p.

g) Derive a criterion involving n and p for determining the expected number of tests with the composite plan is less than the number of tests required with individual testing. [Hint: By “criterion,” I mean a condition that can be checked, such as “composite testing has the smaller number of tests whenever np>10.”]