---
tag: 0AKNPYQ04R
kind: definition
name: Classical definition of probability
aliases: [uniform-probability-on-a-finite-set]
msc: [60A05]
defines: 051CYS71H5
uses: [0ZYBZ34RT2, 0BBTH3YSZA]
sources:
  - url: https://en.wikipedia.org/wiki/Classical_definition_of_probability
    locator: lead section
---

## Definition

Let $\Omega$ be a nonempty [[finite set|0BBTH3YSZA]] of equally likely outcomes.
The classical probability of an event $A \subseteq \Omega$ is
$$P(A) = \frac{|A|}{|\Omega|},$$
the number of favourable outcomes divided by the number of all outcomes.

## Remarks

This $P$ is a [[probability measure|0VGMEBVW1M]] on the power set of $\Omega$, the uniform probability on $\Omega$. Each outcome has probability $1/|\Omega|$.
For two rolls of a fair die, the probability that the sum is $7$ is $6/36 = 1/6$.
Classical probability turns a probability problem into a counting problem. Binomial coefficients and the rule of product often give the counts.
The definition is circular as a foundation, since "equally likely" already uses probability. It also does not apply to infinite sample spaces.

## Explanations

### L1 human

When all outcomes have the same chance, a probability is a fraction: good cases over all cases.
A fair die has $6$ faces and $3$ of them are even, so the chance of an even number is $3/6 = 1/2$.