Why a 99% accurate test can still be wrong most of the time
An interactive lesson on Bayes' theorem and the base rate fallacy. Move the sliders and watch 10,000 imaginary people get tested.
1. Set up the test
How common the condition is in the group being tested (log scale, 0.01% to 50%).
Of people who have the condition, the share who test positive.
Of people who do not have the condition, the share who test negative.
2. What a positive result means
chance you have it, given a positive test (PPV)
chance you are clear, given a negative test (NPV)
of 10,000 people test positive
3. See all 10,000 people
True positive
False positive
False negative
True negative
| Test positive | Test negative | Total | |
|---|---|---|---|
| Has condition | |||
| No condition | |||
| Total | 10,000 |
4. The formula behind it
Bayes' theorem turns "how often does the test catch the condition" into "how likely is the condition, given the test result".
P(condition | positive) = sensitivity x prevalence / [ sensitivity x prevalence + (1 - specificity) x (1 - prevalence) ]
The key insight: when a condition is rare, the healthy group is huge. Even a small false positive rate applied to a huge group produces more false alarms than there are real cases. That is the base rate fallacy: ignoring how common something is before the test.
5. Check yourself
Sources and notes
All scenarios on this page are hypothetical and computed exactly from the slider values; they do not describe any real test. The method follows standard definitions:
- Altman DG, Bland JM. Diagnostic tests 2: Predictive values. BMJ 1994;309:102.
- Gigerenzer G, Hoffrage U. How to improve Bayesian reasoning without instruction: frequency formats. Psychological Review 1995;102(4):684-704.
- Eddy DM. Probabilistic reasoning in clinical medicine: problems and opportunities. In: Kahneman, Slovic, Tversky (eds), Judgment under Uncertainty, 1982.