Definition:Independent Events/Pairwise Independent

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 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 and only 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:

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

holds whenever $\card J = 2$.

Also see

• Pairwise Independence does not imply Independence: although independence (in the general sense) implies pairwise independence, a collection of events can be pairwise independent without being independent.

• Results about independent events can be found here.

Sources

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