Two positive tests for a rare condition

Question

A condition affects 2\% of people. A test catches 90\% of people who have it (sensitivity) and correctly clears 80\% of people who do not, so it false-alarms on 20\% of the healthy. A person takes the test twice, and the two tests are independent given their true status. Both come back positive. What is the probability the person has the condition?

Accepted Answer

Imagine 10{,}000 people: 200 have the condition, 9{,}800 do not. Because the two tests are independent given status, multiply the per-test rates. - Has condition, both positive: 200 	imes 0.90 	imes 0.90 = 162. - Healthy, both positive: 9{,}800 	imes 0.20 	imes 0.20 = 392. Of the 162 + 392 = 554 people with two positives, 162 truly have the condition: P(	ext{condition} \mid ++) = \frac{162}{554} \approx 0.292 = 29\%. A single positive would give only about 10\%, but the second independent positive multiplies the evidence and pushes belief up to roughly 29\%. Takeaway. Independent tests m…

Two positive tests for a rare condition

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