# Definition:Event

## Definition

Let $\EE$ be an experiment.

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

Events are usually denoted $A$, $B$, $C$, and so on.

$U$ is often used to denote an event which is certain to occur.

Similarly, $V$ is often used to denote an event which is impossible to occur.

### Occurrence

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

Let $A, B \in \Sigma$ be events, 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.

### 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.

## Also known as

An event is also known as a random event.

## 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:

$A$ is the event that there is at least one head
$B$ is the event that there are two heads.

### 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.