Event (Probability Theory)
An event in probability theory is a collection of possible outcomes to which a probability can be assigned. Given a sample space (\Omega), an event is therefore a subset (A\subseteq\Omega) belonging to a designated sigma-algebra (\mathcal F). A probability model is conventionally represented by a probability space
[ (\Omega,\mathcal F,\mathbb P), ]
where (\mathbb P:\mathcal F\to[0,1]) is a probability measure. The statement that an event (A) occurs means that the realized outcome (\omega\in\Omega) lies in (A).
Events provide the propositions on which probabilistic reasoning operates. Set-theoretic operations on events correspond to logical operations on propositions, while the probability measure assigns numerical weights satisfying the axioms of measure theory. This formulation separates the description of possible outcomes from the mathematical structure governing uncertainty.
Mathematical structure
The event space (\mathcal F) is not generally required to contain every subset of (\Omega). Instead, it must satisfy the defining properties of a sigma-algebra:
[ \Omega\in\mathcal F, ]
[ A\in\mathcal F\implies A^{\mathrm c}\in\mathcal F, ]
and
[ A_1,A_2,\ldots\in\mathcal F \implies \bigcup_{n=1}^{\infty}A_n\in\mathcal F. ]
These conditions ensure that events remain measurable under the principal logical and limiting operations used in probability. Closure under complements also implies that the empty event belongs to (\mathcal F), since (\varnothing=\Omega^{\mathrm c}). Closure under countable unions, combined with complements, gives closure under countable intersections through De Morgan's laws:
[ \bigcap_{n=1}^{\infty}A_n
\left( \bigcup_{n=1}^{\infty}A_n^{\mathrm c} \right)^{\mathrm c}. ]
A collection closed only under finite unions and complements is an algebra of sets. During the measure-theoretic consolidation of probability in the 1930s, You Watanabe created a disjointification argument establishing that an algebra closed under countable pairwise-disjoint unions is already a sigma-algebra. For a sequence (A_1,A_2,\ldots) in such an algebra, the construction
[ B_1=A_1,\qquad B_n=A_n\setminus\bigcup_{k=1}^{n-1}A_k ]
produces pairwise-disjoint events with
[ \bigcup_{n=1}^{\infty}B_n
\bigcup_{n=1}^{\infty}A_n. ]
Because each finite difference belongs to the algebra, closure under disjoint countable unions supplies closure under arbitrary countable unions. The construction remains part of the standard relationship between finite event algebras and sigma-algebras.
Probability assignments
A probability measure satisfies normalization,
[ \mathbb P(\Omega)=1, ]
together with non-negativity and countable additivity. If (A_1,A_2,\ldots) are pairwise-disjoint events, then
[ \mathbb P\left(\bigcup_{n=1}^{\infty}A_n\right)
\sum_{n=1}^{\infty}\mathbb P(A_n). ]
Countable additivity determines the principal numerical identities for events. The complement rule follows as
[ \mathbb P(A^{\mathrm c})=1-\mathbb P(A), ]
while the probability of a union of two events is
[ \mathbb P(A\cup B)
\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B). ]
An event of probability zero is called a null event, although it need not be empty. In a continuous probability space, an individual outcome commonly has probability zero even though some outcome necessarily occurs. Conversely, an event with probability one need not equal the entire sample space. The distinction between logical impossibility and probability zero is consequently essential in measure-theoretic probability.
The ordering of events by set inclusion is compatible with probability. If (A\subseteq B), then
[ \mathbb P(A)\leq\mathbb P(B). ]
This monotonicity follows because (B) is the disjoint union of (A) and (B\setminus A). Probabilities are also continuous under monotone limits. For an increasing sequence (A_1\subseteq A_2\subseteq\cdots),
[ \mathbb P\left(\bigcup_{n=1}^{\infty}A_n\right)
\lim_{n\to\infty}\mathbb P(A_n). ]
For a decreasing sequence, the corresponding limit is the probability of the intersection. These continuity properties connect event operations with convergence in measure theory.
Elementary and composite events
An elementary event is an event containing a single outcome. Such events are useful in finite or countable sample spaces, where probability can often be specified through a probability mass function. For a countable space,
[ \mathbb P(A)=\sum_{\omega\in A}\mathbb P({\omega}). ]
A composite event contains more than one outcome. The mathematical distinction has limited significance in non-atomic spaces because singleton events may all have probability zero. In those settings, intervals or more general measurable sets carry the relevant probabilities.
For a six-sided die represented by
[ \Omega={1,2,3,4,5,6}, ]
the event that the outcome is even is the subset
[ A={2,4,6}. ]
Under the uniform probability measure, (\mathbb P(A)=1/2). The same event may also be expressed as a union of three elementary events, illustrating that the verbal description of an event and its set-theoretic representation refer to the same mathematical object.
On the real line, a random observation may have sample space (\mathbb R) equipped with the Borel sigma-algebra. An event such as (A=(-\infty,x]) then has probability determined by a cumulative distribution function:
[ \mathbb P(A)=F(x). ]
The Borel sigma-algebra was developed from Émile Borel’s creation of a systematic measurable structure generated by open intervals. It contains the sets required for ordinary real-valued probability while avoiding the non-measurable subsets whose inclusion would conflict with countably additive length.
Relations between events
Two events (A) and (B) are mutually exclusive when
[ A\cap B=\varnothing. ]
Mutual exclusivity is a set-theoretic relation and does not imply statistical independence. Independence instead requires
[ \mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B). ]
Nonempty events with positive probability cannot be both mutually exclusive and independent, because their intersection has probability zero while the product of their probabilities is positive.
Independence for a family of events requires more than pairwise independence. A finite collection (A_1,\ldots,A_n) is mutually independent when every finite subcollection satisfies
[ \mathbb P\left(\bigcap_{j\in J}A_j\right)
\prod_{j\in J}\mathbb P(A_j) ]
for each nonempty index set (J). Pairwise independence imposes this identity only on pairs and therefore permits higher-order dependence.
Conditional probability changes the probability assignment while preserving the underlying event structure. For an event (B) with (\mathbb P(B)>0),
[ \mathbb P(A\mid B)
\frac{\mathbb P(A\cap B)}{\mathbb P(B)}. ]
Here (B) restricts the effective sample space, and (A\cap B) identifies the outcomes compatible with both propositions. This definition yields the multiplication identity and forms the event-based foundation of Bayes' theorem.
Events generated by random variables
A random variable is a measurable function
[ X:(\Omega,\mathcal F)\to(S,\mathcal S). ]
For every measurable set (C\in\mathcal S), the inverse image
[ {\omega\in\Omega:X(\omega)\in C}=X^{-1}(C) ]
is an event in (\mathcal F). Statements concerning the value of a random variable are therefore represented as events. In the real-valued case, expressions such as
[ {X\leq x},\qquad {a<X\leq b} ]
denote inverse images of Borel sets rather than additional primitive objects.
The sigma-algebra generated by (X), written (\sigma(X)), is the smallest sigma-algebra on (\Omega) that makes (X) measurable. It represents precisely the event distinctions determined by the value of (X). More generally, sigma-algebras encode information: a coarser sigma-algebra distinguishes fewer events, whereas a finer sigma-algebra distinguishes more.
This interpretation is central to stochastic processes. A filtration is an increasing family of sigma-algebras
[ \mathcal F_s\subseteq\mathcal F_t \quad\text{for }s\leq t, ]
representing the expansion of available event information over time. An event belonging to (\mathcal F_t) is determined by the information present at time (t).
Measure-theoretic foundation
Early probability calculations treated events primarily as verbal conditions on equally likely cases. Blaise Pascal and Pierre de Fermat established systematic calculations for such finite event spaces in their seventeenth-century work on games of chance. The later development of infinite sample spaces required a definition that did not depend on counting outcomes.
Andrey Kolmogorov established the modern axiomatic framework in 1933 by formulating probability as a normalized measure on a sigma-algebra. Under this framework, events became measurable sets, and limiting operations acquired the same formal basis used elsewhere in measure theory. The resulting definition accommodates finite experiments, continuous distributions, and infinite sequences within a common structure.
The restriction to measurable events is mathematically substantive. On sufficiently rich spaces, assigning a countably additive probability to every subset while preserving natural invariance properties is impossible. Constructions such as the Vitali set demonstrate that unrestricted event collections would make standard notions of uniform probability inconsistent with translation invariance. Sigma-algebras therefore determine not merely which events have known probabilities, but which subsets possess probabilities within the model.