Solution
The Rare Disease Test
Show the problem again
A disease affects 1% of a population. A test is 99% sensitive (it detects the disease when present) and 99% specific (it is negative when the disease is absent). You test positive. What is the probability you actually have the disease?
Worked solution
The answer is 50%. By Bayes' theorem with a population of 10,000: 100 have the disease and 99 test positive; 9,900 don't and 99 falsely test positive. Of the 198 positives, exactly half are real: P(disease | positive) = 99/198 = 1/2. The base rate drags the answer far below the intuitive 99%: the canonical base-rate-neglect example.
Source: Base-rate neglect in medical testing; the canonical framing follows Casscells, Schoenberger and Grayboys (1978). Statement written for AxiomIQ.