Question

during the initial months of the covid-19 pandemic, resources for testing samples for the disease were scarce. what if specimens were pooled and several samples were tested at once? if the pool tested negative, then all of the individuals who submitted specimens in the sample are free from the disease. if the pool tested positive, then at least one of the specimens in the pool was positive, and all of the specimens in the pool would be tested individually. in the long run, does this procedure result in fewer tests?
assume that 5% of specimens that are tested for covid-19 yield positive results. (the actual rate was much higher when testing was not extensive, but overall, when testing is being done extensively, 5% is a reasonable estimate.)
suppose that 1 million people present themselves for testing and that specimens are pooled together in groups of size 5. what is the expected number of tests that would have to be performed?
calculate the expected total number of tests that would have to be performed.
expected total number of tests:
tests

Explanation:

Step1: Determine number of pools

There are 1,000,000 people, and specimens are pooled in groups of 5. So the number of pools is $\frac{1000000}{5} = 200000$.

Step2: Calculate probability a pool is negative

The probability that a single specimen is negative is $1 - 0.05 = 0.95$. For a pool of 5 (all negative for the pool to be negative), the probability a pool is negative is $0.95^5$. Calculate $0.95^5 \approx 0.77378$.

Step3: Probability a pool is positive

The probability a pool is positive is $1 - 0.77378 = 0.22622$.

Step4: Expected number of pool tests

We start with 200000 pool tests.

Step5: Expected number of individual tests

For positive pools, we test all 5 specimens. The expected number of positive pools is $200000 \times 0.22622 \approx 45244$. So the expected number of individual tests is $45244 \times 5 = 226220$.

Step6: Total expected tests

Total expected tests = pool tests + individual tests = $200000 + 226220 = 426220$.

Answer:

426220