by Eliezer Yudkowsky Feb 22 2016 updated Oct 1 2016

20% of patients have Diseasitis. 90% of sick patients and 30% of healthy patients turn a tongue depressor black. You turn a tongue depressor black. What's the chance you have Diseasitis?

[summary: A nurse is screening a set of patients for an illness, Diseasitis, using a tongue depressor that tends to turn black in the presence of the disease.

The probability that a patient with a blackened tongue depressor has Diseasitis can be found using Bayes' rule.]

A nurse is screening a student population for a certain illness, Diseasitis (lit. "inflammation of the disease").

You are testing for the presence of the disease using a color-changing tongue depressor with a sensitive chemical strip.

One of your students comes into the office, takes your test, and turns the tongue depressor black.

Given only that information, what is the probability that they have Diseasitis?

This problem is used as a central example in several introductions to Bayes's Rule, including all paths in the Arbital Guide to Bayes' Rule and the High-Speed Intro to Bayes' Rule. A simple, unnecessarily difficult calculation of the answer can be found in Frequency diagrams: A first look at Bayes.