Calculate posterior probabilities with Bayes theorem. Visualize prior, likelihood, and posterior with tree diagrams and area plots.
Enter a prior probability, a true-positive rate (sensitivity), and a false-positive rate, and the calculator plugs them into Bayes' theorem to return the posterior probability of the event given a positive observation. It shows the same numbers two ways: a branching probability tree with prior x likelihood along each path, and a stacked area plot where dragging the prior slider shows how the posterior share shifts in real time. The most common surprise is how low the posterior stays when the prior is small, even with a highly accurate test - the base rate dominates.
A concrete input and expected output from the current implementation.
Input
Prior P(event) = 1%, sensitivity P(positive | event) = 90%, false-positive rate P(positive | no event) = 5%
Expected output
P(event | positive) = 15.38%
P(positive) = 0.9x0.01 + 0.05x0.99 = 0.0585, so the posterior is 0.009 / 0.0585 = 0.1538, meaning most positives come from the much larger pool of non-events even though the test is 90% sensitive.
The tool takes the prior P(A), the likelihood P(B|A), and the false-positive rate P(B|not A), then computes P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|not A)P(not A)]. It displays the joint probabilities for each of the four prior/likelihood combinations alongside the final ratio, so the numerator and full denominator are both visible, not just the result.
The tree diagram branches first on the prior (event / no event), then on the observation (positive / negative) under each branch, labeling every path with its joint probability. The area plot stacks the same four joint probabilities as proportional bands across a 0-1 axis, and both redraw immediately as any of the three input fields changes.
This is the base-rate effect: when the prior is small, the absolute number of false positives from the large 'no event' group can exceed the true positives from the small 'event' group, even at high sensitivity. The tool's tree view makes this visible by showing both false-positive and true-positive path probabilities side by side.
Only if the two pieces of evidence are conditionally independent given the event and given its complement - the standard 'naive Bayes' assumption. In that case, re-entering the posterior as the new prior and running a second pass with the next likelihood gives the correct combined posterior. If the two observations are dependent (the second is more or less likely given that the first already occurred), that second likelihood must instead be conditioned on the first observation, which this single-pass calculator does not model - chaining runs naively will produce the wrong answer.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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