Axioms of Probability¶

Axiom¶

Let $S$ be a sample space, let $\mathcal{F}$ be a sigma-algebra of events, that is, a collection of events containing $S$ and closed under complements and countable unions, and let $\mathbb{P}\colon\mathcal{F}\to\mathbb{R}$ $R$ . The function $\mathbb{P}$ satisfies the axioms of probability if:

$\mathbb{P}(A)\ge 0$ for every event $A$ .
$\mathbb{P}(S)=1$ $1$ .
If $A_1,A_2,\ldots$ are pairwise disjoint events, then

\mathbb{P}\left(\bigcup_{i=1}^{\infty}A_i\right)=\sum_{i=1}^{\infty}\mathbb{P}(A_i).

Interpretation¶

The axioms are the minimal rules that every assignment of probabilities must obey: probabilities are never negative, the whole sample space has probability one, and probabilities of disjoint events add up. Every other rule, such as the complement rule in the Key Results, follows from them.

Three panels for the three axioms: an event A inside the sample space with nonnegative probability, the whole sample space S with probability 1, and two disjoint events A and B whose probabilities add.

The three panels match the three axioms. An event $A$ inside $S$ gets a number $\mathbb{P}(A)\ge0$ $0$ ; the whole of $S$ gets $1$ , since some outcome is always realized; and two events $A$ and $B$ without overlap give $\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)$ $P (B)$ . The figure draws axiom 3 for two events, while the axiom asks it for any sequence of pairwise disjoint events.

The first two axioms fix the scale: with additivity they give $\mathbb{P}(A)+\mathbb{P}(A^c)=\mathbb{P}(S)=1$ $1$ , so no probability exceeds $1$ . Additivity needs disjointness: on a fair die $\mathbb{P}(\{1,2\})+\mathbb{P}(\{2,3\})=\tfrac{4}{6}$ $\frac{4}{6}$ , but $\mathbb{P}(\{1,2,3\})=\tfrac{3}{6}$ $\frac{3}{6}$ , because the outcome $2$ is counted twice. Countably many events are needed for infinite sample spaces: when a fair coin is tossed until the first head, "a head eventually appears" is the disjoint union of "first head at toss $k$ " for $k=1,2,\ldots$ $1, 2, \dots$ , with probability $\tfrac{1}{2}+\tfrac{1}{4}+\cdots=1$ $1$ , a geometric series.

The axioms do not say which probabilities fit a given experiment. "All outcomes equally likely", as in naive probability, is an extra modelling assumption that satisfies them. A function that satisfies the axioms is called a probability function.

Key Results¶

From the axioms, $\mathbb{P}(A^c)=1-\mathbb{P}(A)$ for every event $A$ .
If $A\subseteq B$ $B$ , then $\mathbb{P}(A)\le \mathbb{P}(B)$ $P (B)$ .
Finite additivity follows from countable additivity: for pairwise disjoint $A_1,\ldots,A_n$ take $A_{n+1}=A_{n+2}=\cdots=\varnothing$ in axiom 3 and use $\mathbb{P}(\varnothing)=0$ $0$ , which gives $\mathbb{P}(A_1\cup\cdots\cup A_n)=\mathbb{P}(A_1)+\cdots+\mathbb{P}(A_n)$ $P (A_{n})$ .

Examples¶

In a fair die model, nonnegativity assigns no negative chance to any face, and the sample space has probability one.
If the events 'roll a one' and 'roll a two' are disjoint, their union has probability equal to the sum of their probabilities.
For nested events such as 'roll at most two' and 'roll at most four', monotonicity gives the smaller event no larger probability.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 1

Aliases: Kolmogorov axioms, probability axioms, axioms of Kolmogorov, axiomatic definition of probability Symbols: infinity, real numbers, subset, sum, union

Last updated: 20 October 2026 03:02