A positive screen with an imperfect test

Question

A disease affects 1\% of a population. A screening test catches 80\% of true cases (sensitivity) and correctly clears 90\% of healthy people (specificity), so it false-alarms on 10\% of the healthy. A randomly selected person tests positive. What is the probability that they actually have the disease?

Accepted Answer

Use natural frequencies on 10{,}000 people: - Sick: 1\% 	imes 10{,}000 = 100 people; of these, 80\% = 80 test positive (true positives). - Healthy: 9{,}900 people; of these, 10\% = 990 test positive (false positives). Among the 80 + 990 = 1{,}070 positives, only 80 are truly sick: P(	ext{disease} \mid +) = \frac{80}{1{,}070} \approx 0.075 = 7.5\%. Even after a positive result, the person is about 92\% likely to be healthy. The tiny sick pool simply cannot outweigh the flood of false positives thrown off by the large healthy pool, especially with only 80\% sensitivity. In odds form: prior od…

A positive screen with an imperfect test

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