Posterior disease probability after a positive screen

Question

A disease affects $0.5\%$ of a population. A test always comes back positive for someone who has the disease, but it also returns a positive result for $7\%$ of healthy people. A randomly chosen person tests positive. How likely is it that they actually have the disease?

Short answer

About $6.7\%$ . By Bayes' rule the probability is $\dfrac{0.005}{0.005+0.995\times0.07}=\dfrac{0.005}{0.07465}\approx0.067$ . False positives from the large healthy group swamp the true positives.

Adapted from Crack, Heard on the Street, Q4.30, p.43; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
Conditional Probability, Total Probability, and Bayes' Rule
Sample Spaces, Events, and the Probability Axioms

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