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.