# Probability

**Type:** Concept  
**Domain:** Probability & Statistics  
**Codex URL:** /mathematics/probability-statistics/probability/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Mathematical framework for chance and uncertainty. Formalised by Kolmogorov's axioms (1933). Origins in the Pascal-Fermat correspondence (1654).

## Kolmogorov's Axioms
1. P(A) ≥ 0
2. P(S) = 1
3. P(A∪B) = P(A)+P(B) for mutually exclusive A, B

## Conditional probability & independence
P(A|B) = P(A∩B)/P(B). Independent if P(A|B)=P(A).

## Bayes' Theorem
P(A|B) = P(B|A)P(A)/P(B) — Thomas Bayes, 1763 (posthumous)

## Interpretations
Frequentist (long-run frequency) vs Bayesian (degree of belief) — active philosophical debate, same mathematical axioms.

## Sources
### Tier 1
- Kolmogorov, A.N. (1933). *Grundbegriffe der Wahrscheinlichkeitsrechnung*.
- Feller, W. (1968). *An Introduction to Probability Theory*. Vol. 1, 3rd ed.
### Tier 2
- Hacking, I. (2006). *The Emergence of Probability*. 2nd ed.
### Tier 3
- Devlin, K. (2008). *The Unfinished Game*. Basic Books.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*