Bayes’ Theorem Calculator
Update a prior probability with new evidence using Bayes’ theorem — ideal for medical-test, spam-filter and diagnostic reasoning.
How it works
P(A|B) = [ P(B|A) · P(A) ] / [ P(B|A) · P(A) + P(B|¬A) · (1 − P(A)) ]
Bayes’ theorem revises a prior probability P(A) once you observe evidence B. The denominator is the total probability of seeing that evidence — the true-positive path plus the false-positive path — which is why a very accurate test can still yield a low posterior when the condition is rare. This base-rate effect is the reason a 99% accurate screen for a 1-in-100 condition still leaves most positives as false alarms.
Worked example
With a base rate of 1%, a 99% true-positive rate and a 5% false-positive rate: numerator = 0.99 × 0.01 = 0.0099; denominator = 0.0099 + 0.05 × 0.99 = 0.0594; so P(A|B) = 0.0099 / 0.0594 = 16.67%.
Sources: Bayes’ theorem — Wikipedia
Frequently asked questions
Why is the posterior so low when the test is 99% accurate?
Because the condition is rare. With a 1% base rate there are far more healthy people, so even a small 5% false-positive rate produces more false positives than true positives. Bayes’ theorem weighs both, giving about 17%.
What is the false positive rate?
It is P(B|not A) — the chance the test comes back positive for someone who does not have the condition. It equals one minus the test’s specificity.
Can I chain a second test?
Yes. Use the posterior from the first test as the prior for the second. Two independent positives multiply the odds, which is why confirmatory testing works.
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This calculator is for educational purposes. Double-check important results before acting on them.