Bayes' Theorem Calculator

Bayes' theoremupdates a prior belief about a hypothesis A once you observe evidence B. It rests on two simpler rules of probability and combines them into a single formula. Throughout, let A be the event of interest (“has the condition”, “is spam”) and B the observed evidence (“tests positive”, “contains the word”).

The full statement, with the denominator expanded, is:

P(A|B) = P(B|A)·P(A) / [ P(B|A)·P(A) + P(B|¬A)·P(¬A) ]

1. Complement the prior.P(¬A) = 1 − P(A). This is the basic complement axiom of probability (Grinstead & Snell).
2. Find the evidence by total probability. The chance of seeing B at all splits across the two ways it can happen — when A is true and when A is false:

   P(B) = P(B|A)·P(A) + P(B|¬A)·P(¬A)

   This marginal P(B) is the normalising constant. Source: NIST/SEMATECH e-Handbook; Grinstead & Snell §4.
3. Apply Bayes' theorem. Divide the true-positive term by the evidence to get the posterior P(A|B) = P(B|A)·P(A) / P(B). This is the same rule scikit-learn states for Naive Bayes, P(y|x) = P(y)·P(x|y) / P(x).
4. Complement the posterior. P(¬A|B) = 1 − P(A|B), the probability that A is false despite the evidence.

The calculator above also recomputes the posterior a second, independent way — the odds form. The prior odds P(A)/P(¬A) are multiplied by the likelihood ratio P(B|A)/P(B|¬A) to give the posterior odds, which convert back to a probability via odds / (1 + odds). Because this uses a ratio of ratios rather than the normalisation above, agreement between the two routes is a genuine correctness check, and the four joint probabilities of the tree are confirmed to sum to 1. When a likelihood is 0 the evidence can vanish (P(B) = 0); the tool then reports the posterior as undefined rather than dividing by zero.

A key intuition the tool makes visible: the posterior depends heavily on the base rate. A highly sensitive test applied to a rare condition still produces mostly false positives, because the large healthy population contributes many false alarms while the small affected population contributes few true ones. That is the false-positive paradox, and the probability tree shows exactly where the numbers come from.

• scikit-learn — Naive Bayes (states Bayes' theorem and its use in classification)
• NIST/SEMATECH e-Handbook of Statistical Methods — conditional probability and total-probability rules
• Grinstead & Snell — Introduction to Probability (open access, AMS) — Bayes' theorem derivation

The formulas on this page were last cross-checked against these sources on 2026-06-11. Bayes' theorem is a stable mathematical identity, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled.