Bayes' Theorem Calculator
Calculate conditional probability with Bayes' theorem from prior, likelihood, and evidence.
Includes a worked medical test sensitivity and specificity example.
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)) ≈ 8.7% This counterintuitive result shows how low prevalence reduces positive predictive value.
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. This iterative process is Bayesian inference — fundamental to machine learning, spam filters, medical diagnosis, and 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.
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) ≈ 94%. 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.
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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