Bayes' Theorem Calculator

Bayes’ Theorem P(A|B) = P(B|A) × P(A) / P(B) Or equivalently: Posterior = (Likelihood × Prior) / Evidence Named after Thomas Bayes (England, c. 1763), refined by Pierre-Simon Laplace.

Terms Explained P(A) = Prior probability: probability of A before observing B. P(B|A) = Likelihood: probability of observing B given that A is true. P(A|B) = Posterior: probability of A being true after observing B. P(B) = Evidence (marginal probability): P(B) = P(B|A)×P(A) + P(B|¬A)×P(¬A)

Medical Test Example A test for a disease with: Prevalence (prior): P(disease) = 1% → P(A) = 0.01 Sensitivity (true positive rate): P(positive | disease) = 95% → P(B|A) = 0.95 Specificity: P(negative | no disease) = 90% → P(B|¬A) = 1 - 0.90 = 0.10 Result: P(disease | positive test) = (0.95 × 0.01) / ((0.95 × 0.01) + (0.10 × 0.99)) ≈ 8.76% This counterintuitive result shows how low prevalence reduces positive predictive value.

The same example in whole people, which is much easier to see. Take 10,000 people. About 100 have the disease and 9,900 do not. Of the 100 sick, the test catches 95. Of the 9,900 healthy, it wrongly flags 990. So 1,085 people test positive and only 95 of them are actually ill, which is 8.76%. Nothing about that calculation needs Bayes’ theorem written out. Counting people is the same arithmetic in a form the reader can check on their fingers, and the result panel shows both.

Bayesian Updating Each new piece of evidence updates the probability: New posterior = P(A|new evidence) from current prior. The prior of the next calculation becomes the previous posterior. That iterative process is Bayesian inference, and it is the engine under machine learning, spam filters, medical diagnosis and ordinary scientific reasoning.

Bayes Factor BF = P(B|H₁) / P(B|H₀) The ratio of likelihoods for two hypotheses. BF > 3: moderate evidence for H₁. BF > 10: strong evidence. Note what the Bayes factor does not include: the prior. It measures only how much the evidence should move you, not where you end up, which is why a test with a huge Bayes factor can still leave a rare disease unlikely.

A second example, spam filtering. Say 20% of incoming mail is spam, and the word “lottery” turns up in 60% of spam but only 1% of legitimate mail. An email containing “lottery” is spam with probability (0.60 × 0.20) / (0.60 × 0.20 + 0.01 × 0.80) ≈ 93.75%. Chaining many word-level probabilities like this is exactly how a naive Bayes spam classifier scores a whole message.

The trap to avoid: base rate neglect. The most common mistake in everyday probability reasoning is ignoring the prior. Shown a “99% accurate” test for a rare disease, most people blurt out “99% chance you have it” and forget how rare the disease is to begin with. Bayes’ theorem is the antidote: it forces the base rate and the test result to be combined, instead of reading the test result on its own.