Bayes' Theorem

Bayes' theorem is the rule for flipping a conditional probability around: it turns P(evidence given hypothesis), which you can often measure, into P(hypothesis given evidence), which is what you actually want to know. It combines that measurement with how common the hypothesis was to begin with (the prior).

It is the formal recipe for updating beliefs with data and the foundation of Bayesian (貝氏) statistics. Its most counter-intuitive lesson is the base-rate effect: for a rare condition, even an accurate test produces mostly false alarms, because there are so many more healthy people to mistakenly flag.