# Bayes' Theorem

Canonical: https://manyspace.aporia.systems/topics/bayes-theorem-mth-prob-199
Updated: 2026-10-13

Mathematics · Probability · MTH-PROB-199

- Difficulty: 3 / 5
- Importance: 5 / 5
- Study time: 4–7.5 h

total probability, prior/posterior, diagnostic testing, base rates

## Why it matters

Reverses conditional probabilities — the basis of diagnosis, spam filters and Bayesian statistics.

## Where it is used

Spam filtering, medical diagnostics, A/B test interpretation, Bayesian ML.

## How you know you have learned it

Solve ≥10 representative problems with ≥80% adjusted accuracy.

## What you need first

- [Conditional Probability & Independence](https://manyspace.aporia.systems/topics/conditional-probability-independence-mth-prob-198) (required)
- [Integration by Parts](https://manyspace.aporia.systems/topics/integration-by-parts-mth-calc2-140) (required)
- [Counting Principles](https://manyspace.aporia.systems/topics/counting-principles-mth-disc-112) (helpful)

## What it opens

- [Bayesian Statistics](https://manyspace.aporia.systems/topics/bayesian-statistics-mth-stats-222)
- [Bayesian Reasoning](https://manyspace.aporia.systems/topics/bayesian-reasoning-aiml-prb-005)

## Resources

- An Introduction to Probability Theory (Vol 1) — William Feller
- Introduction to Probability (Blitzstein & Hwang) — Blitzstein & Hwang
- Introduction to Probability (Blitzstein & Hwang) — Blitzstein & Hwang