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A probability provides a quantitative description of the likely occurrence of a particular event.

Overview

Probability is conventionally expressed on a scale from $0$ to $1$: a rare or unlikely event has a probability close to $0$ while a common or expected event has a probability close to $1$.

The notion of probability has been shaped by Andrey Kolmogorov and his Axioms of Probability. For a sample space $\Omega$ and a sigma algebra $S$, a probability function $P$ satisfies

  1. $P(A) \geq 0$ for all $A \in S$.
  2. $P(\Omega) = 1$.
  3. When $A_1,A_2,... \in S$ are pairwise disjoint, $P\left( \cup_{i=1}^{\infty} A_i \right) = \sum_{i=1}^{\infty}P(A_i).$

These axioms form a set of useful rules for calculating probabilities.

The probability of an event has been interpreted variously as its long-run relative frequency and as a personal degree of belief (subjective probability).

References

The following threads on math.se contain references to resources about probability:

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