Definition:Independent Events/General Definition

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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.


Pairwise Independent

Let $\mathcal A = \left\{{A_i: i \in I}\right\}$ be a set of events of $\mathcal E$.


Then $\mathcal A$ is pairwise independent iff:

$\forall j, k \in I: \Pr \left({A_j \cap A_k}\right) = \Pr \left({A_j}\right) \Pr \left({A_k}\right)$

That is, if every pair of events of $\mathcal E$ are independent of each other.


That is, $\mathcal A$ is pairwise independent iff the condition for general independence:

$\displaystyle \Pr \left({\bigcap_{i \mathop \in J} A_i}\right) = \prod_{i \mathop \in J} \Pr \left({A_i}\right)$

holds whenever $\left|{J}\right| = 2$.


Sources