Bayes' Theorem Calculator
Update a prior probability with new evidence — the engine of rational belief, spam filters and medical reasoning.
Bayes' theorem is how to update belief with evidence, and its lesson is the base-rate effect: with a 1% prior, even a 99%-accurate test (5% false positives) yields only ~17% posterior probability. Evidence multiplies the prior — it doesn't replace it. Ignoring the base rate is the most common reasoning error in statistics.
Formula
About Bayes' Theorem Calculator
Bayes' theorem is the mathematics of updating beliefs with evidence, and it powers everything from spam filters to medical diagnosis to rational decision-making. This calculator takes a prior probability (the base rate), a true-positive rate and a false-positive rate, and computes the posterior — your updated belief after seeing the evidence. Its most important lesson is the base-rate effect: with a low prior, even very strong evidence yields a surprisingly modest posterior, because evidence multiplies the prior rather than replacing it. Neglecting the base rate is the single most common error in probabilistic reasoning.
How to use Bayes' Theorem Calculator
- 1Enter your values into Bayes' Theorem Calculator — sensible, domain-typical defaults are pre-filled so you see a real result immediately.
- 2The result recomputes live using the formula shown on the page; there is no button to press.
- 3Adjust any input to compare scenarios, then read the worked example to see the substituted numbers.
Why use Bayes' Theorem Calculator?
- ✓Computes Bayes' Theorem instantly in your browser — no sign-up, no upload, no server round-trip.
- ✓100% free and unlimited, with the exact formula shown: P(H|E) = P(E|H).
- ✓Runs entirely client-side, so every value you enter stays private on your device.
- ✓Live recompute as you type, with a worked example and authoritative references for trust.
Frequently asked questions
Why does strong evidence give a weak posterior here?+
The base rate dominates. With a 1% prior, the rare true cases are vastly outnumbered by the common non-cases, so even a small false-positive rate generates more false alarms than true detections. A 99%-accurate test on a 1%-prevalence condition yields only ~17% posterior — the famous base-rate neglect that fools doctors, jurors and analysts alike.
What's the difference between prior, likelihood and posterior?+
The prior P(H) is your belief before evidence (the base rate). The likelihood P(E|H) is how probable the evidence is if the hypothesis is true. The posterior P(H|E) is your updated belief after seeing the evidence. Bayes combines them: posterior is proportional to likelihood times prior, normalized over all hypotheses.
How is this the same as the PPV/base-rate calculator?+
Mathematically identical — PPV (positive predictive value) IS the posterior probability of disease given a positive test. Bayes' theorem is the general engine; the medical PPV framing is one application. Spam filtering (P(spam|words)), fault diagnosis and A/B-test interpretation are others. Learn it once here and it applies everywhere evidence updates belief.
How do I update on multiple pieces of evidence?+
Apply Bayes repeatedly: today's posterior becomes tomorrow's prior. If evidence pieces are conditionally independent, you can multiply their likelihood ratios. This sequential updating is exactly how naive Bayes classifiers combine many word-features, and how rational reasoning accumulates evidence over time — each new fact refines, rather than resets, your belief.
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