Last updated: 2026-06-23

Bayes’ Theorem

Bayes’ theorem is a rule for updating a probability when new evidence arrives. It takes a starting belief about how likely something is, combines it with how strongly the evidence points one way or another, and returns a revised belief. This calculator applies it to the most common case — a diagnostic test — turning a condition’s prevalence and a test’s accuracy into the probability that a positive result is real.

Conditional probability

A conditional probability $P(A|B)$ is the probability that $A$ is true given that $B$ has been observed. It differs from the plain probability $P(A)$ because the observation of $B$ carries information. A positive medical test does not make a disease certain; it shifts the probability upward by an amount that depends on both how good the test is and how common the disease was to begin with. The related Conditional Probability and Bayes' Theorem Calculator calculator works directly with $P(A \cap B)$ and $P(B)$ when those are known.

The formula

Bayes’ theorem relates the two conditional directions:

$P(A|B) = \dfrac{P(B|A)\,P(A)}{P(B)}$

Each term has a name:

• The prior $P(A)$ is the probability of $A$ before the evidence — for a test, the prevalence of the condition.
• The likelihood $P(B|A)$ is how probable the evidence is when $A$ holds — the test’s sensitivity, or true-positive rate.
• The marginal (or evidence) $P(B)$ is the overall probability of seeing the evidence at all, across cases where $A$ is true and where it is not. It expands to $P(B|A)\,P(A) + P(B|\lnot A)\,P(\lnot A)$, where $P(B|\lnot A)$ is the false-positive rate.
• The posterior $P(A|B)$ is the result: the probability of $A$ after the evidence.

The base-rate fallacy

The most common mistake in reading a test result is to confuse the test’s accuracy with the probability that a positive is real. These are different questions, and when the condition is rare they have very different answers. A test that is 90% sensitive and 95% specific sounds reliable, but if only 1 in 100 people has the condition, the small false-positive rate is applied to the 99 healthy people — and there are so many of them that false alarms outnumber true positives several times over. Ignoring the prevalence in this way is called the base-rate fallacy.

Worked example

Suppose a condition affects 1% of the tested population, the test catches 90% of true cases, and it falsely flags 5% of healthy people. The prior is a prevalence of 0.01, the sensitivity is 0.90, and the false-positive rate is 0.05:

So a positive result raises the probability of the condition from 1% to roughly 15%. The test moved the needle a lot — fifteenfold — yet a single positive still leaves the result more likely false than true. This is why rare-condition screening is usually followed by a second, more specific test rather than treated as conclusive.

Extensions

The same theorem can be written in odds form, which often makes the update easier to reason about: the posterior odds equal the prior odds times the likelihood ratio $P(B|A) / P(B|\lnot A)$. For the example above the prior odds are one to ninety-nine and the likelihood ratio is eighteen, giving posterior odds of eighteen to ninety-nine, or about 15% — the same answer. The closely related Odds Ratio Calculator compares two such ratios directly.

Bayes’ theorem also supports sequential updating: after one test, the posterior becomes the prior for the next. Two independent positive tests, each applied in turn, push the probability far higher than either alone — which is exactly how confirmatory testing earns its confidence. Whenever evidence accumulates piece by piece, the same rule applies at each step, with yesterday’s conclusion serving as today’s starting point.