Definition:Event

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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:

$A$ 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.


Sources