# Definition:Independent Events/General Definition

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## 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 $I$:

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

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.

### Pairwise Independent

Let $\mathcal A = \family {A_i}_{i \mathop \in I}$ 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, if every pair of events of $\EE$ are independent of each other.

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

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

holds whenever $\card J = 2$.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.7$: Independent Events: $(22)$