# Event (probability theory)
In probability theory, an event is a subset of the sample space of a random experiment, that is, a set of outcomes to which a probability can be assigned. For a die roll with sample space \( \Omega = \{1,2,3,4,5,6\} \), the subset \( \{2,4,6\} \) is the event "an even number is thrown". Formally, events are the elements of a σ-algebra \( \mathcal F \) on \( \Omega \), and a probability measure \( \mathbb P \) assigns each event a number in \( [0,1] \); the triple \( (\Omega, \mathcal F, \mathbb P) \) is called a probability space.[1](https://few.vu.nl/~molenaar/courses/StatR/chapters/C-10-probability_theory.html) • [2](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
| Key fact | Detail |
|---|---|
| Definition | An event is a subset of the sample space \( \Omega \); probabilities are defined on events, not on individual outcomes.[1](https://few.vu.nl/~molenaar/courses/StatR/chapters/C-10-probability_theory.html) |
| Formal home | Events are the elements of a σ-algebra \( \mathcal F \subseteq \mathcal P(\Omega) \), the domain of the probability measure \( \mathbb P \).[2](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) |
| Closure rules | An event space contains \( \Omega \) and \( \emptyset \) and is closed under complements and countable unions.[3](https://ee.stanford.edu/~gray/sp.pdf) |
| Simple and compound events | A simple event consists of one outcome, such as \( \{6\} \) for a die; a compound event contains several, such as \( \{2,4,6\} \).[1](https://few.vu.nl/~molenaar/courses/StatR/chapters/C-10-probability_theory.html) |
| Extreme events | \( \Omega \) is the certain event ("something happens") and \( \emptyset \) the impossible event ("nothing happens").[3](https://ee.stanford.edu/~gray/sp.pdf) |
| Discrete case | For a countable sample space one usually takes \( \mathcal F = \mathcal P(\Omega) \); a discrete space with \( n \) elements has \( 2^n \) events.[2](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) • [3](https://ee.stanford.edu/~gray/sp.pdf) |
| Continuous case | For uncountable \( \Omega \), such as \( [0,1] \), some subsets cannot be assigned a probability consistently, so events must be restricted to a σ-algebra such as the Borel sets.[4](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) |
## Events and the probability space
The modern framework defines an experiment by three objects. The sample space \( \Omega \) collects all possible outcomes; the σ-algebra \( \mathcal F \) collects the events whose probabilities can be measured; and the probability measure \( \mathbb P \) assigns to every event \( A \in \mathcal F \) a number \( \mathbb P(A) \in [0,1] \), the probability that \( A \) occurs.[2](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) • [5](https://faculty.washington.edu/yenchic/26A_stat516/Lec1_probability.html) The measure satisfies three axioms: \( \mathbb P(\Omega) = 1 \), \( \mathbb P(A) \ge 0 \) for every event, and countable additivity, \( \mathbb P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mathbb P(A_i) \) for pairwise disjoint events.[5](https://faculty.washington.edu/yenchic/26A_stat516/Lec1_probability.html)
**Why probabilities attach to sets, not outcomes.** For many experiments the quantities of interest are not single outcomes. A glucose measurement has sample space the real numbers, and the useful statement is that the concentration falls in the healthy range \( [4,6] \) mM, an event with positive probability, while any single exact value has probability zero in a continuous model.[1](https://few.vu.nl/~molenaar/courses/StatR/chapters/C-10-probability_theory.html) When outcomes are equally likely, the probability of an event reduces to counting: for two independent fair coin tosses with \( \Omega = \{HH, HT, TH, TT\} \), \( \mathbb P(A) = |A|/4 \) for every subset \( A \).[2](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
## The event space, or σ-algebra
Not every subset of \( \Omega \) need be an event. The collection of events is required to be a σ-algebra (also called a σ-field or event space), meaning it satisfies closure conditions that keep the calculus of events internally consistent:[3](https://ee.stanford.edu/~gray/sp.pdf)
1. \( \Omega \in \mathcal F \), and with it \( \emptyset \), the certain and impossible events.[3](https://ee.stanford.edu/~gray/sp.pdf)
2. If \( A \in \mathcal F \), then its complement \( A^c \in \mathcal F \), so "not A" is always an event.[3](https://ee.stanford.edu/~gray/sp.pdf)
3. If \( A_1, A_2, \ldots \in \mathcal F \), then \( \bigcup_{i=1}^{\infty} A_i \in \mathcal F \), so "at least one of these events" is an event; countable intersections follow by [De Morgan's laws](https://www.edgepedia.org/de-morgans-laws).[3](https://ee.stanford.edu/~gray/sp.pdf) • [6](https://cni.iisc.ac.in/courses/random/Probability%20Foundations/lecture-01.html)
A collection satisfying only complementation and finite unions is called a field or algebra of sets; the countable-union condition is what distinguishes a σ-algebra.[3](https://ee.stanford.edu/~gray/sp.pdf) The smallest event space on any \( \Omega \) is the trivial one, \( \{\emptyset, \Omega\} \), which carries no information about the outcome; the largest is the power set \( \mathcal P(\Omega) \), containing every subset.[4](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) • [3](https://ee.stanford.edu/~gray/sp.pdf) Between these extremes, the event space can be tuned to the question at hand: for a die, the collection \( \{\emptyset, \Omega, \{1,3,5\}, \{2,4,6\}\} \) is a valid σ-algebra that distinguishes only even from odd rolls.[1](https://few.vu.nl/~molenaar/courses/StatR/chapters/C-10-probability_theory.html)
**Generated event spaces.** Given a family \( \mathcal G \) of subsets that must be events, the generated σ-algebra \( \sigma(\mathcal G) \) is the smallest event space containing every set in \( \mathcal G \). A single event \( A \) generates \( \{\emptyset, A, A^c, \Omega\} \).[6](https://cni.iisc.ac.in/courses/random/Probability%20Foundations/lecture-01.html) On the real line, the σ-algebra generated by the open intervals \( (a,b) \) is the Borel σ-algebra \( \mathcal B(\mathbb R) \), which also contains all open sets; it is the standard event structure for continuous outcomes.[7](https://www.tropp.caltech.edu/notes/Tro24-Probability-Theory-LN.pdf)
## Why not always the power set
For finite or countable sample spaces the power set works and is the usual choice.[2](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) For uncountable spaces the situation changes. When \( \Omega = [0,1] \), there exist subsets of \( \Omega \) that cannot be assigned a probability in a way that is consistent with the axioms, so it is too much to demand probabilities on all subsets; the events must live in some σ-algebra \( \mathcal F \subseteq 2^{\Omega} \).[4](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) Uncountable sample spaces are unavoidable in practice, arising in infinite coin-tossing and in any experiment modeled by a continuous distribution such as the normal.[4](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) The power set is therefore too large to be useful for continuous spaces, and constructions such as the Borel σ-algebra supply the working middle ground.[3](https://ee.stanford.edu/~gray/sp.pdf) • [7](https://www.tropp.caltech.edu/notes/Tro24-Probability-Theory-LN.pdf)
## Events and random variables
A random variable is a measurable function \( X : \Omega \to \mathbb R \): for every [Borel set](https://www.edgepedia.org/borel-set) \( B \), the preimage \( X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B\} \) must lie in \( \mathcal F \).[8](https://descartes.manoa.hawaii.edu/ee342/morea/discrete-rv/experience-measurable.html) Measurability is exactly the condition that turns statements about \( X \) into events: \( X^{-1}(B) \) is an event, so \( \mathbb P(X \in B) \) is defined. If \( X \) is not measurable, some Borel set \( B \) has \( X^{-1}(B) \notin \mathcal F \), and that statement about \( X \) cannot be assigned a probability.[8](https://descartes.manoa.hawaii.edu/ee342/morea/discrete-rv/experience-measurable.html)
## Historical note
The event-based axiomatic framework descends from Andrei Nikolaevich Kolmogorov's monograph *Grundbegriffe der Wahrscheinlichkeitsrechnung*, published in 1933, which put probability's modern mathematical formalism in place by building the theory on measure theory, with events as the measurable sets and probability as a normalized measure.[9](https://arxiv.org/abs/1802.06071) • [10](https://mathematik.com/Kolmogorov/) The measure-theoretic grounding resolved paradoxes that had troubled earlier treatments and remains the standard mathematical answer to what a probability is: a measure \( P \) on a sample space that assigns total measure 1 to the whole space.[11](https://stat210a.berkeley.edu/fall-2026/reader/probability.html)
How the number \( \mathbb P(A) \) should be interpreted is a separate question from its definition. In the frequentist view, \( \mathbb P(A) \) is the relative frequency with which \( A \) occurs in the limit of many independent repetitions of the experiment; in the Bayesian view, it quantifies degree of belief that \( A \) will occur or is true. The two interpretations attach to the same mathematical object, and the disagreements between the corresponding statistical schools concern how the formalism connects to real-world questions, not the construct itself.[12](https://stat238.berkeley.edu/spring-2026/lecturesix238spring2026/) • [11](https://stat210a.berkeley.edu/fall-2026/reader/probability.html)
## References
1. [Introduction to probability theory, Statistics with R (VU Amsterdam)](https://few.vu.nl/~molenaar/courses/StatR/chapters/C-10-probability_theory.html)
2. [Probability spaces and Random Variables, Carnegie Mellon University lecture notes](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
3. [Robert M. Gray, An Introduction to Signal Detection and Estimation, chapter 2: Probability spaces](https://ee.stanford.edu/~gray/sp.pdf)
4. [Clare Wallace, Axioms of probability, Probability 1 lecture notes, Durham University](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html)
5. [Yen-Chi Chen, Lecture 1: Review on Probability and Statistics, STAT 516, University of Washington](https://faculty.washington.edu/yenchic/26A_stat516/Lec1_probability.html)
6. [Sample and Event Space, Probability and Random Processes, Indian Institute of Science](https://cni.iisc.ac.in/courses/random/Probability%20Foundations/lecture-01.html)
7. [Joel A. Tropp, Probability Theory and Stochastic Processes, Caltech CMS 117 lecture notes](https://www.tropp.caltech.edu/notes/Tro24-Probability-Theory-LN.pdf)
8. [Measurable functions, EE 342, University of Hawaii at Manoa](https://descartes.manoa.hawaii.edu/ee342/morea/discrete-rv/experience-measurable.html)
9. [The origins and legacy of Kolmogorov's Grundbegriffe, arXiv:1802.06071](https://arxiv.org/abs/1802.06071)
10. [Kolmogorov: Foundations of the Theory of Probability](https://mathematik.com/Kolmogorov/)
11. [Probability, STAT 210A course reader, UC Berkeley](https://stat210a.berkeley.edu/fall-2026/reader/probability.html)
12. [Aditya Guntuboyina, Interpretation of Probability, STAT 238, UC Berkeley](https://stat238.berkeley.edu/spring-2026/lecturesix238spring2026/)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Probability spaces and axioms › Modeling experiments and events*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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