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Bayes’ theorem: an intuitive explanation

How new evidence updates a probability without confusing P(A|B) with P(B|A).

Bayes updates the probability of a hypothesis $A$ after evidence $B$ : $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}.$

The three parts

$P(A)$ is the prior probability; $P(B\mid A)$ measures how compatible the evidence is with the hypothesis; $P(B)$ normalizes across every way the evidence can occur.

An example with 1,000 people

Suppose 10 people have a condition. A test identifies 9 of them but is also positive for 99 healthy people. Among 108 positive tests, only 9 are true cases, so the updated probability is $9/108\approx8.3\%$ , not $90\%$ .

The common mistake

$P(B\mid A)$ and $P(A\mid B)$ are not interchangeable. Test sensitivity among people with a condition is not the probability of the condition after a positive result.

Frequently asked questions

Why does the prior matter so much?

For a rare event, false positives can outnumber true positives even when a test is fairly accurate.

Is Bayes only for medical tests?

No. It is used in statistics, diagnosis, machine learning, and any setting where evidence updates an initial belief.