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, for all finite subsets $J$ of $I$:
 * $\displaystyle \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$

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