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

Based on prior epidemiological studies, you expect that around 20% of the students in a screening population like this one will actually have Diseasitis.

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

Among students with Diseasitis, 90% turn the tongue depressor black.
30% of the students without Diseasitis will also turn the tongue depressor black.

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.