Event¶

Definition¶

Let $S$ be a sample space. An event is a subset $A\subseteq S$ whose occurrence can be assigned a probability.

Interpretation¶

An event is a yes-or-no statement about the result of an experiment, written as the set of outcomes for which the answer is yes. It occurs exactly when the realized outcome belongs to it.

The sample space of a die roll with the even numbers 2, 4 and 6 highlighted as the event that the roll is even.

In the figure, "the roll is even" is the subset $A=\{2,4,6\}$ of $S=\{1,2,3,4,5,6\}$ ${1, 2, 3, 4, 5, 6}$ : a roll of $4$ makes $A$ occur, a roll of $3$ does not, and with equally likely faces $\mathbb{P}(A)=\tfrac{3}{6}=\tfrac{1}{2}$ $\frac{1}{2}$ .

Words become set operations: " $A$ or $B$ " is the union, " $A$ and $B$ " the intersection, "not $A$ " the complement. The empty set never occurs and $S$ itself always occurs.

A common slip is to confuse the outcome $4$ with the event $\{4\}$ : only events, which are sets, are given probabilities.

Key Results¶

Events are subsets of the sample space.
Events can be combined using union, intersection, and complement.
A probability function assigns probabilities to events, not to informal statements outside the model.

Examples¶

In a die roll, the event 'roll an even number' is the set of outcomes two, four, and six.
In a reliability model, the event 'system fails before noon' is a subset of possible histories.
In a survey, the event 'respondent is under thirty' collects all outcomes with that property.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 1

Keywords: random event Symbols: subset

Last updated: 20 October 2026 03:02