Track 4 · Probability · Lesson 6 of 9
Bayes' theorem
Use Bayes' theorem and natural frequencies to turn P(evidence | hypothesis) into P(hypothesis | evidence), and explain why base rates matter.
A screening test for a silent condition is described as “92% sensitive and 95% specific”. A person tests positive. Most people — including many professionals — guess they very probably have the condition. When the condition is rare, the truth is closer to one in four. Bayes' theorem explains why, and it applies just as much to fraud alerts, quality alarms and spam filters.
Start with natural frequencies — imagine 10,000 people, prevalence 2%, sensitivity 92%, specificity 95%:
- 200 have the condition. The test finds 92% of them: 184 true positives.
- 9,800 do not. The test wrongly flags 5% of them: 490 false positives.
- Altogether 674 people test positive, and only 184 of them have the condition: .
The test is good, but healthy people outnumber ill people 49 to 1, so even a small false-positive rate produces more false positives than true ones.
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Compute a positive or negative predictive value with Bayes' theorem or natural frequencies
- Explain the base-rate fallacy and why PPV falls when a condition is rare
- Tell P(A | B) from P(B | A) and spot the prosecutor's fallacy
- Update a probability step by step as new, independent evidence arrives
- Check Bayes' theorem against an observed two-way table