Mathematics
What is Bayes' theorem?
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.
See it, don’t just read it.
Watch a 2-minute lesson with voice + animation that explains bayes' theorem.
Key things to understand
- 1Written P(A|B) = P(B|A) × P(A) / P(B): the chance of A given evidence B.
- 2P(A) is the 'prior' (before evidence); P(A|B) is the 'posterior' (after evidence).
- 3It formalizes common sense: rare conditions stay unlikely even after a positive-but-imperfect test.
- 4Used in spam filters, medical diagnosis, machine learning, and courtroom statistics.
Frequently asked questions
- 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.