Definition:Independent Events/General Definition

Definition
Let $\mathcal E$ be an experiment with probability space $\left({\Omega, \Sigma, \Pr}\right)$. Let $\mathcal A = \left\{{A_i: i \in I}\right\}$ be a family of events of $\mathcal E$.

Then $\mathcal A$ is independent iff, for all finite subsets $J$ of $I$:
 * $\displaystyle \Pr \left({\bigcap_{i \mathop \in J} A_i}\right) = \prod_{i \mathop \in J} \Pr \left({A_i}\right)$

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