Definition:Independent Events

Definition

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 $\map \Pr B > 0$.

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:

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

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:

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

The definition can be made to apply to more than just two events.

Let $\AA = \family {A_i}_{i \mathop \in I}$ be an indexed family of events of $\EE$.

Then $\AA$ is independent if and only if, for all finite subsets $J$ of $I$:

$\ds \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$

That is, if and only if the occurrence of any finite collection of $\AA$ has the same probability as the product of each of those sets occurring individually.

If $A$ and $B$ are not independent, then they are dependent (on each other), and vice versa.

Event Independence is Symmetric: thus it makes sense to refer to two events as independent of each other.