Definition:Independent Events/Definition 1

Definition

Let $\mathcal E$ be an experiment with probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Let $A, B \in \Sigma$ be events of $\mathcal E$ such that $\Pr \left({A}\right) > 0$ and $\Pr \left({B}\right) > 0$.

The events $A$ and $B$ are defined as independent (of each other) iff the occurrence of one of them does not affect the probability of the occurrence of the other one.

Formally, $A$ is independent of $B$ iff:

$\Pr \left({A \mid B}\right) = \Pr \left({A}\right)$

where $\Pr \left({A \mid B}\right)$ denotes the conditional probability of $A$ given $B$.