Bayes' Theorem Calculator

Frequently Asked Questions

What is Bayes' theorem?

P(A|B) = P(B|A)·P(A) / P(B), where P(B) = P(B|A)·P(A) + P(B|¬A)·P(¬A). It updates a prior belief with new evidence to a posterior.

Why can a positive test still mean low risk?

With a rare condition (low prior), even a good test produces many false positives relative to true positives, so P(disease | positive) can be surprisingly small.

What are sensitivity and false-positive rate?

Sensitivity = P(positive | has condition); false-positive rate = P(positive | no condition) = 1 − specificity. Both feed the posterior calculation.

What is the natural-frequency restatement?

Expressing the same problem as "out of 1,000 people…" makes it intuitive: count true vs false positives directly instead of juggling conditional probabilities.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

Bayes' theorem updates a prior belief in light of new evidence. With A the event of interest and B the observed evidence, P(A | B) = [P(B | A) × P(A)] ÷ P(B), where the denominator expands by the law of total probability: P(B) = P(B | A)P(A) + P(B | ¬A)P(¬A). In diagnostic-test language, P(A) is the prevalence (prior), P(B | A) is the sensitivity (true-positive rate), and P(B | ¬A) is the false-positive rate (1 − specificity). The output P(A | B) is the posterior - the probability you actually have the condition given a positive result. This is an exact algebraic computation, not an approximation; only your input precision limits accuracy.

Worked Example: Rare Disease, Accurate Test

Prevalence P(A) = 1% = 0.01, sensitivity P(B | A) = 99%, false-positive rate P(B | ¬A) = 5%. Then P(B) = 0.99 × 0.01 + 0.05 × 0.99 = 0.0099 + 0.0495 = 0.0594. The posterior is P(A | B) = 0.0099 ÷ 0.0594 ≈ 0.1667 - only about 17%. Despite a 99%-sensitive test, a positive result more often than not is a false alarm, because the disease is rare. This is the famous base-rate fallacy: most clinicians intuitively guess ~95%.

The Natural-Frequency Restatement

The same logic is far clearer in counts. Imagine 10,000 people. With 1% prevalence, 100 truly have the disease and 9,900 do not. The test catches 99 of the 100 true cases (99% sensitivity) but also flags 5% of 9,900 healthy people = 495 false positives. Of the 99 + 495 = 594 positive tests, only 99 are genuine - again about 17%. Presenting Bayes' theorem this way dramatically improves correct reasoning, which is why this calculator shows both the formula and the frequency view.

When to Use It & Common Mistakes

Medical screening: interpreting a positive mammogram, HIV, or COVID test against prevalence.

Spam filters & fraud: probability an email is spam given a flagged keyword.

Forensics & law: avoiding the prosecutor's fallacy - P(evidence | innocent) ≠ P(innocent | evidence).

The dominant mistake is ignoring the base rate: a highly accurate test on a rare condition still yields mostly false positives. Confusing P(B | A) with P(A | B) is the same error in symbolic form.

Interpretation & Limitations

The posterior is only as good as the prior - garbage prevalence in, garbage answer out. After a positive result the posterior becomes the new prior for any follow-up test (chaining Bayes), which is why a second independent positive raises confidence sharply. The model assumes the sensitivity and false-positive rate are accurate and that the test result is conditionally independent given disease status; correlated repeat tests violate this. Inputs must be probabilities between 0 and 1 (use 0.05, not 5). The arithmetic here is exact to displayed precision.

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