# Experiment/Examples

## Examples of Experiments

### 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:

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

where $\mathrm H$ denotes heads and $\T$ 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: $\Sigma = \set {\forall \omega \in \Omega: \omega > 4}$.

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