Bayes' Theorem

Posterior probability with a tree diagram

Probability of the hypothesis before testing (0–1)

True-positive rate — chance of evidence when H is true (0–1)

Chance of evidence when H is false (0–1)

P(H|E) = P(E|H)·P(H) / [ P(E|H)·P(H) + P(E|¬H)·P(¬H) ]
Posterior P(H|E)
0.1538 (15.38%)
Evidence P(E)
0.0585 (5.85%)
Posterior given no evidence P(H|¬E)
0.0011
True-positive path P(H ∩ E)
0.009
False-positive path P(¬H ∩ E)
0.0495
Bayes' TheoremProbability tree splitting the prior into the true-positive and false-positive paths that produce the evidence, with the resulting posterior0.010.990.90.10.050.95H∩E0.009H∩¬E0.001¬H∩E0.0495¬H∩¬E0.9405•H¬HP(H|E) = 0.1538

About this calculator

The Bayes' Theorem Calculator finds the posterior probability P(H|E) from a prior, a sensitivity (true-positive rate), and a false-positive rate. It draws a probability tree showing how the prior splits into the true-positive path P(H ∩ E) and the false-positive path P(¬H ∩ E), reports the total evidence P(E), and updates the belief in the hypothesis. A clear way to see the base-rate fallacy in medical screening, spam filters, and diagnostic testing.