Bayes' Theorem Calculator

P(H | E) = P(E | H) · P(H) / P(E), where P(E) = P(E | H) · P(H) + P(E | ¬H) · P(¬H).

P(H) — prior, your belief before seeing the evidence (the base rate).

P(E | H) — likelihood, how often the evidence appears when the hypothesis is true (sensitivity for a medical test).

P(E | ¬H) — false-positive rate, how often the evidence appears when the hypothesis is false.

P(H | E) — posterior, the updated probability after the evidence.

A disease has a 1% prevalence (P(H) = 0.01). A test has 99% sensitivity (P(+ | H) = 0.99) and 5% false-positive rate (P(+ | ¬H) = 0.05).

A random patient tests positive. What's the chance they actually have the disease?

P(+) = 0.99·0.01 + 0.05·0.99 = 0.0099 + 0.0495 = 0.0594.

P(H | +) = (0.99·0.01) / 0.0594 ≈ 0.1667 — only 17%. Even with a 99% accurate test, a positive on a rare condition is more likely a false alarm than the disease.

This is why population screening is hard: when base rates are low, false positives dominate the positives.

Bayes is a multiplier on the prior odds. Prior odds = P(H) / P(¬H); posterior odds = prior odds × likelihood ratio (LR+ = sensitivity / FPR).

If LR+ = 20, a positive multiplies your prior odds by 20. A 1% prior (odds 1:99) becomes posterior odds 20:99 → P ≈ 0.168.

This explains why repeating an independent positive test helps: each positive multiplies the odds again. Two independent positives at LR+ = 20 give posterior odds 1:99 · 20² = 400:99 → P ≈ 0.80.

Spam filters — P(spam | features). Prior = base rate of spam in your inbox; likelihood = how often those words appear in spam vs ham.

A/B significance — P(real lift | observed difference). Frequentist p-values implicitly assume a 50/50 prior, which is rarely reasonable for already-shipped features.

Forensics, fault detection, search-and-rescue — anywhere prior knowledge plus an imperfect signal needs to combine into a posterior probability.

Base-rate neglect — focusing on the test's accuracy and ignoring the prevalence. The example above is the classic example.

Inverting conditionals — P(E | H) is not P(H | E). A symptom may be 95% likely given the disease, but the disease may still be rare given the symptom.

Assuming independence of tests when they share systematic errors. Two correlated tests don't multiply odds as cleanly as the formula suggests.