# Definition:Experiment

## Definition

An experiment can be defined in natural language as a test to see what happens.

### Informal Definition

An experiment is defined as:

a course of action whose consequence is not predetermined.

### Formal Definition

An experiment, which can conveniently be denoted $\EE$, is a measure space $\struct {\Omega, \Sigma, \Pr}$ such that $\map \Pr \Omega = 1$.

## Also known as

An experiment is also known as:

a trial

or:

an observation.

## Examples

### Throwing a $6$-Sided Die

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

The sample space of $\EE$ is $\Omega = \set {1, 2, 3, 4, 5, 6}$.
Various events can be identified:
$(1): \quad$ The result is $3$:
The event space of $\EE$ is: $\Sigma = \set 3$.
$(2): \quad$ The result is at least $4$:
The event space of $\EE$ is: $\Sigma = \set {\forall \omega \in \Omega: \omega > 4}$.
$(3): \quad$ The result is a prime number:
The event space of $\EE$ is: $\Sigma = \set {2, 3, 5}$.
The probability measure is defined as:
$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 6$

### Tossing 2 Coins

Let $\EE$ be the experiment of 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.

Suppose we are interested only in whether the coins fall alike ($\mathrm A$) or different ($\mathrm D$).

Then the sample space of $\EE$ is: $\Sigma = \set {\forall \omega \in \Omega: \omega > 4}$.

$\Omega = \set {\mathrm A, \mathrm D}$