In the summer of 2020, the U.S. was considering pooled testing of COVID-19. This problem explores the math behind pooled testing. Since the availability of tests is limited, the testing center proposes the following pooled testing technique: Two samples are randomly selected and combined. The combined sample is tested. If the combined sample tests negative, then both people are assumed negative. If the combined sample tests positive, then both people need to be retested for the disease. Suppose in a certain population that 5% of the people being tested for COVID-19 actually have COVID-19 and assume that tests are 100% accurate at detection (i.e. they perfectly determine whether a person has COVID-19 or not). Let X be the total number of tests that are run in order to test two randomly selected people. a) What is the pmf of X for: P(X=1) = P(X=2) = P(X=3) = b) What is the expected value of X? c) Imagine that instead three samples are combined, and let Y be the total number of tests that are run in order to test three randomly selected people. If the pooled test is positive, then all three people need to be retested individually. What is the expected value of Y? d) If your only concern is to minimize the expected number of tests given to the population, which technique would you recommend? Choice 1 of 3:Standard testing (one test per person, no combining) Choice 2 of 3:Pooled testing with two samples combined Choice 3 of 3:Pooled testing with three samples combined.
In the summer of 2020, the U.S. was considering pooled testing of COVID-19. This problem explores the math behind pooled testing. Since the availability of tests is limited, the testing center proposes the following pooled testing technique:
- Two samples are randomly selected and combined. The combined sample is tested.
- If the combined sample tests negative, then both people are assumed negative.
- If the combined sample tests positive, then both people need to be retested for the disease.
Suppose in a certain population that 5% of the people being tested for COVID-19 actually have COVID-19 and assume that tests are 100% accurate at detection (i.e. they perfectly determine whether a person has COVID-19 or not). Let X be the total number of tests that are run in order to test two randomly selected people.
a) What is the pmf of X for:
P(X=1) =
P(X=2) =
P(X=3) =
b) What is the
c) Imagine that instead three samples are combined, and let Y be the total number of tests that are run in order to test three randomly selected people. If the pooled test is positive, then all three people need to be retested individually.
What is the expected value of Y?
d) If your only concern is to minimize the expected number of tests given to the population, which technique would you recommend?
Choice 1 of 3:Standard testing (one test per person, no combining)
Choice 2 of 3:Pooled testing with two samples combined
Choice 3 of 3:Pooled testing with three samples combined.
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