Category:Definitions/Independent Events

This category contains definitions related to Independent Events.
Related results can be found in Category:Independent Events.

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr A > 0$ and $\map \Pr B > 0$.

Definition 1

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

Formally, $A$ is independent of $B$ if and only if:

$\condprob A B = \map \Pr A$

where $\condprob A B$ denotes the conditional probability of $A$ given $B$.

Definition 2

The events $A$ and $B$ are defined as independent (of each other) if and only if the occurrence of both of them together has the same probability as the product of the probabilities of each of them occurring on their own.

Formally, $A$ and $B$ are independent if and only if:

$\map \Pr {A \cap B} = \map \Pr A \map \Pr B$