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 $\EE$.


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.


Prime Number on $6$-Sided Die

Let $\EE$ be the experiment of throwing a standard $6$-sided die.

The sample space of $\EE$ is $\Omega = \set {1, 2, 3, 4, 5, 6}$.

Consider the subset $E \subseteq \Omega$ defined as:

$E = \set {2, 3, 5}$

Then $E$ is the event that the result of $\EE$ is a prime number.


Even Number on $6$-Sided Die

Let $\EE$ be the experiment of throwing a standard $6$-sided die.

The sample space of $\EE$ is $\Omega = \set {1, 2, 3, 4, 5, 6}$.

Consider the subset $E \subseteq \Omega$ defined as:

$E = \set {2, 4, 6}$

Then $E$ is the event that the result of $\EE$ is even.


Score Divisible by $3$ on $6$-Sided Die

Let $\EE$ be the experiment of throwing a standard $6$-sided die.

Let $E \subseteq \Omega$ be the event that the result of $\EE$ is divisible by $3$.

The sample space of $\EE$ is $\Omega = \set {1, 2, 3, 4, 5, 6}$.

Hence:

$E = \set {3, 6}$

and so:

$\map \Pr E = \dfrac 1 3$


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.


Also see

  • Results about events can be found here.


Sources

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