# Definition:Classical Probability Model

## Definition

Let $\EE$ be an experiment

The classical probability model on $\EE$ is a mathematical model that defines the probability space $\struct {\Omega, \Sigma, \Pr}$ of $\EE$ as follows:

$(1) \quad$ All outcomes of $\EE$ are equally likely
$(2) \quad$ There are a finite number of outcomes.

Then:

$\map \Pr {\text {event occurring} } := \dfrac {\paren {\text {number of outcomes favorable to event} } } {\paren {\text {total number of outcomes possible} } }$

or formally:

$\map \Pr \omega := \dfrac {\card \Sigma} {\card \Omega}$

where:

$\card {\, \cdot \,}$ denotes the cardinality of a set
$\omega$ denotes an event: $\omega \in \Omega$
$\Sigma$ denotes the event space: $\Sigma \subseteq \Omega$
$\Omega$ denotes the sample space.

## Historical Note

The classical probability model arose when games of chance were first analyzed in the $17$th century, in the context of the gambling games of the European nobility.

Because of its assumption of equiprobable outcomes, this model is particularly useful when used in this context.