Why Your Positive Test Is Probably Wrong
A positive result on a screening test sounds alarming, but with a rare disease and an imperfect test, most positive results can be false alarms. This works out how base rate and test accuracy combine to answer a question people usually get wrong: given a positive test, what is the actual chance of having the disease?
Introduction
Doctors, patients and statisticians regularly confuse P(A given B) with P(B given A): for a rare disease these can be very different numbers. This exploration works through a screening-test scenario with a chosen disease prevalence and test accuracy, using tree diagrams and two-way tables to keep the two conditional probabilities straight. The natural extension is Bayes' theorem, which is AA HL content and not part of the AI syllabus, so an AI student should build and read the tree diagram directly rather than citing the theorem by name.
Guiding Questions
- Choose a disease and find, or reasonably estimate, its prevalence and a screening test's false positive and false negative rates.
- Build a tree diagram or a two-way table for 10,000 people using those numbers.
- From your diagram, calculate P(disease | positive test). How does it compare with the test's advertised accuracy?
- Vary the prevalence. At what point does a positive result stop being worth worrying about?
- Explain your result in one paragraph a non-mathematician could follow, without using the phrase 'conditional probability'.