\begin{definition}[Definition:Independent Events/General 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''' {{iff}}, 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, {{iff}} the occurrence of any finite collection of $\AA$ has the same probability as the product of each of those sets occurring individually.
\end{definition}