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What is Bayes' theorem?

MathematicsLesson on requestFree to read

Quick answer

Bayes' theorem is a formula for updating the probability of a belief when new evidence arrives. It combines how likely the evidence is under your hypothesis with how likely the hypothesis was to begin with, producing a revised, better-informed probability.

Key things to understand

  1. Written P(A|B) = P(B|A) × P(A) / P(B): the chance of A given evidence B.
  2. P(A) is the 'prior' (before evidence); P(A|B) is the 'posterior' (after evidence).
  3. It formalizes common sense: rare conditions stay unlikely even after a positive-but-imperfect test.
  4. Used in spam filters, medical diagnosis, machine learning, and courtroom statistics.

People also ask

Why can a positive medical test still likely be wrong?
If a disease is rare, most positives come from the large healthy group's false positives — Bayes' theorem shows the true-positive chance can stay low.
What is a prior?
Your probability estimate before seeing the new evidence. Bayes' theorem updates it into a 'posterior' once evidence arrives.
Where is Bayes' theorem used?
Spam detection, diagnostics, A/B testing, robotics, and most modern machine-learning and statistics.