---
tag: 0DAX8PMAJ4
kind: concept
name: Probability axioms
aliases: [kolmogorov-axioms]
msc: [60A05]
uses: [0W7NFW23RF, 00SQGJNRTB, 04WSW8T3SY]
sources:
  - url: https://en.wikipedia.org/wiki/Probability_axioms
    locator: Kolmogorov axioms
same_as:
  wikidata: Q974605
  wikipedia: Probability axioms
---

## Summary

The probability axioms of Kolmogorov are the three rules that a probability function $P$ on the events of a sample space $\Omega$ must obey.
First, $P(A) \ge 0$ for each [[event|00SQGJNRTB]] $A$.
Second, $P(\Omega) = 1$.
Third, for each sequence $A_1, A_2, \dots$ of pairwise disjoint events, $P\left(\bigcup_{n} A_n\right) = \sum_{n} P(A_n)$. This is countable additivity.
A function that obeys these rules is a [[probability measure|0VGMEBVW1M]].

## Remarks

Andrey Kolmogorov stated these axioms in 1933.
The other rules of probability follow from the axioms. Examples are $P(\varnothing) = 0$, $P(A) \le 1$, the rule for the complement and the addition rule for two events.

## Explanations

### L1 human

The axioms say three things: a probability is never negative, something always happens, and the probabilities of events that cannot happen together add up.
For a fair coin, $P(\{H\}) = P(\{T\}) = 1/2$, and $P(\{H, T\}) = 1/2 + 1/2 = 1$.