Event (Probability) · aiwiki.page

In probability theory, an event is a set of possible outcomes to which a probability is assigned. It represents a condition on the result of a random experiment: the event occurs precisely when the observed outcome belongs to that set. Formally, an event is a member of the specified collection of measurable subsets of a sample space, rather than necessarily an arbitrary subset. Events are the objects on which the probability measure of a probability space is defined. (ocw.mit.edu)

Outcomes and events

An outcome, usually written $\omega$ , is an individual element of the sample space $\Omega$ . An event $A$ is a subset of $\Omega$ , and its occurrence is expressed by

\omega\in A.

One outcome can belong to several events, so several events can occur in the same experiment. (ocw.mit.edu)

For example, in a model of one roll of a six-sided die,

\Omega=\{1,2,3,4,5,6\}.

The event “the result is even” is $A=\{2,4,6\}$ , while “the result exceeds four” is $B=\{5,6\}$ . An outcome of $6$ belongs to both events. If the die is fair, the equally likely outcome model gives

P(A)=\frac{3}{6}=\frac12, \qquad P(B)=\frac{2}{6}=\frac13.

Counting favorable outcomes gives probabilities only when the underlying outcomes are equally likely. (live.ocw.mit.edu)

An event containing one outcome is often called a simple or elementary event. The distinction between the outcome $\omega$ and its singleton set $\{\omega\}$ is important: the latter is an event only if it belongs to the chosen event collection. (ocw.mit.edu)

Measurability and the event collection

A probability space is a triple

(\Omega,\mathcal F,P),

where $\Omega$ is the sample space, $\mathcal F$ is a sigma-algebra of events, and $P$ assigns probabilities to its members. The collection $\mathcal F$ contains $\Omega$ , is closed under complements, and is closed under countable unions. These properties also imply closure under countable intersections. (live.ocw.mit.edu)

On a finite or countably infinite sample space, a standard choice is the power set $\mathcal P(\Omega)$ , containing every subset. However, smaller sigma-algebras are possible. On uncountable spaces, restricting the event collection is essential in standard continuous models because some subsets cannot be assigned probabilities consistently with the intended measure. (ocw.mit.edu)

For real-valued models, a common choice is the Borel sigma-algebra, generated by open intervals. This measure-theoretic framework specifies which conditions are measurable and therefore eligible to have probabilities. It does not require every conceivable subset to be an event. (ts-server.statlect.com)

Operations on events

Operations from set theory express combinations of conditions. For events $A$ and $B$ :

Event	Meaning
$A\cup B$	At least one of $A$ or $B$ occurs
$A\cap B$	Both occur
$A^c=\Omega\setminus A$	$A$ does not occur
$A\setminus B$	$A$ occurs but $B$ does not

The union uses inclusive “or”: it includes outcomes at which both events occur. The empty set $\varnothing$ is the impossible event, while $\Omega$ is the sure event. (ocw.mit.edu)

Two events are mutually exclusive, or disjoint, when $A\cap B=\varnothing$ . Their logical relationships obey identities such as De Morgan’s laws:

(A\cup B)^c=A^c\cap B^c, \qquad (A\cap B)^c=A^c\cup B^c.

These identities hold independently of the probabilities assigned to the events. (live.ocw.mit.edu)

Probabilities of events

The probability measure satisfies nonnegativity, normalization $P(\Omega)=1$ , and countable additivity: for pairwise disjoint events $A_1,A_2,\ldots$ ,

P\!\left(\bigcup_{n=1}^{\infty}A_n\right) =\sum_{n=1}^{\infty}P(A_n).

Consequently,

P(\varnothing)=0,\qquad 0\leq P(A)\leq1,\qquad P(A^c)=1-P(A).

For arbitrary events,

P(A\cup B)=P(A)+P(B)-P(A\cap B).

The subtraction avoids counting the intersection twice. Also, $A\subseteq B$ implies $P(A)\leq P(B)$ . (ocw.mit.edu)

The event and its probability are different objects. An event describes a set of outcomes; its probability depends on the probability law. The same subset may therefore have different probabilities under different models. (live.ocw.mit.edu)

Conditioning and independence

For an event $B$ with $P(B)>0$ , the conditional probability of $A$ given $B$ is

P(A\mid B)=\frac{P(A\cap B)}{P(B)}.

It describes the probability of $A$ when attention is restricted to outcomes in $B$ . (ocw.mit.edu)

Events $A$ and $B$ are independent when

P(A\cap B)=P(A)P(B).

When $P(B)>0$ , this is equivalent to $P(A\mid B)=P(A)$ . Independence depends on the probability measure, whereas mutual exclusivity is a set-theoretic relationship. Disjoint events with positive probabilities cannot be independent: their intersection has probability zero, but the product of their probabilities is positive. For larger collections, pairwise independence does not generally imply mutual independence. (ocw.mit.edu)

Events and random variables

A random variable assigns a value to each outcome. Statements about its value denote events; for example,

\{X\leq x\} =\{\omega\in\Omega:X(\omega)\leq x\}.

For a real-valued random variable, measurability ensures that these threshold sets are events. The cumulative distribution function is consequently defined by

F_X(x)=P(X\leq x).

Thus expressions such as $P(X=x)$ are shorthand for probabilities of sets of underlying outcomes. (ocw.mit.edu)

Conversely, every event $A$ determines an indicator random variable:

\mathbf1_A(\omega)= \begin{cases} 1,&\omega\in A,\\ 0,&\omega\notin A. \end{cases}

Its expected value satisfies

\mathbb E[\mathbf1_A]=P(A).

This converts questions about event occurrence into questions about numerical random variables and allows event counts to be studied using sums of indicators. (ocw.mit.edu)

Zero probability and certainty

A probability-zero event need not be empty. For example, under the uniform distribution on $[0,1]$ , every singleton has probability zero, although each is a possible modeled outcome. The entire interval nevertheless has probability one. There is no conflict with countable additivity because the interval is an uncountable union of singletons. (ocw.mit.edu)

An event of probability one is said to occur almost surely. This need not mean it equals the entire sample space: it may exclude a nonempty probability-zero event. The distinction separates literal impossibility or certainty, expressed by $\varnothing$ and $\Omega$ , from probability-zero or probability-one statements under a particular measure. (ocw.mit.edu)

References

Discrete Stochastic Processes, Chapter 1: Introduction and Review of Probabilityocw.mit.edu
30 Introduction to Statistical Methods in Economics, Lecture 2live.ocw.mit.edu
Introduction to Probability: Lecture 1: Probability Models and Axiomsocw.mit.edu
Introduction to Probability, Selected Textbook Summary Materialocw.mit.edu
Mathematics for Computer Scienceocw.mit.edu