# Definition:Event

## Contents

## Definition

Let $\EE$ be an experiment.

An **event in $\EE$** is an element of the event space $\Sigma$ of $\EE$.

### Occurrence

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.

Let the outcome of the experiment be $\omega \in \Omega$.

Then the following real-world interpretations of the occurrence of events can be determined:

- If $\omega \in A$, then
**$A$ occurs**.

- If $\omega \notin A$, that is $\omega \in \Omega \setminus A$, then
**$A$ does not occur**.

- If $\omega \in A \cup B$, then
**either $A$ or $B$ occur**.

- If $\omega \in A \cap B$, then
**both $A$ and $B$ occur**.

- If $\omega \in A \setminus B$, then
**$A$ occurs but $B$ does not occur**.

- If $\omega \in A * B$, where $*$ denotes symmetric difference, then
**either $A$ occurs or $B$ occurs, but not both**.

### Simple Event

A **simple event** in $\EE$ is an event in $\EE$ which consists of exactly $1$ elementary event.

That is, it is a singleton subset of the sample space $\Omega$ of $\EE$.

## Also defined as

Some sources define an **event** as **a subset of the sample space $\Omega$**.

However, while this is technically consistent with the definition as given here, it misses the nuance that it is an element of a specified set of such subsets of $\Omega$ that comprise the event space.

## Examples

### Tossing $2$ Coins

Let $\EE$ be the experiment consisting of tossing $2$ coins.

From Tossing $2$ Coins, the sample space of $\EE$ is:

- $\Omega = \set {\tuple {\mathrm H, \mathrm H}, \tuple {\mathrm H, \mathrm T}, \tuple {\mathrm T, \mathrm H}, \tuple {\mathrm T, \mathrm T} }$

where $\mathrm H$ denotes heads and $\mathrm T$ denotes tails.

Let $A$ be the subset of $\Omega$ defined as:

- $A = \set {\tuple {\mathrm H, \mathrm H}, \tuple {\mathrm H, \mathrm T}, \tuple {\mathrm T, \mathrm H} }$

Let $B$ be the subset of $\Omega$ defined as:

- $A = \set {\tuple {\mathrm H, \mathrm H} }$

Then:

### Arbitrary Space

Let $\EE$ be an experiment whose sample space is defined as $\Sigma = \set {e_1, e_2, e_3}$.

The complete set of events of $\EE$ is:

- $\set {\set {e_1}, \set {e_2}, \set {e_3}, \set {e_1, e_2}, \set {e_1, e_3}, \set {e_2, e_3}, \set {e_1, e_2, e_3}, \O}$

The simple events of $\EE$ are:

- $E_1 = \set {e_1}, E_2 = \set {e_2}, E_3 = \set {e_3}$

while $\O$ is the trivial event.

## Sources

- 1965: A.M. Arthurs:
*Probability Theory*... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.2$ Sample spaces and events - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): $\S 4.2$: Trees and Probability - 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.2$: Outcomes and events - 1991: Roger B. Myerson:
*Game Theory*... (previous) ... (next): $1.2$ Basic Concepts of Decision Theory - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**event**