Definition:Independent Events/General Definition

Definition

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

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.

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

Then $\AA$ is pairwise independent if and only if:

$\forall j, k \in I: \map \Pr {A_j \cap A_k} = \map \Pr {A_j} \map \Pr {A_k}$

That is, $\AA$ is pairwise independent if and only if the condition for general independence:

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

holds whenever $\card J = 2$.