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? to begin, calculate the probability that a pool of five specimens yields a positive result. assume that the five specimens are independent. give your answer to at least six decimal places. probability the pool tests positive:

Explanation:

Step1: Define Probability of Negative

The probability that a single specimen is negative is \( 1 - 0.05 = 0.95 \) (since 5% are positive, so 95% are negative).

Step2: Probability All Negative

For a pool of 5 independent specimens, the probability that all are negative (so the pool tests negative) is \( 0.95^5 \) (by multiplication rule for independent events).

Step3: Probability Pool Positive

The probability that the pool tests positive is the complement of all being negative: \( 1 - 0.95^5 \).
Calculate \( 0.95^5 \approx 0.7737809375 \), then \( 1 - 0.7737809375 = 0.2262190625 \approx 0.226219 \).

Answer:

\( 0.226219 \)