Pairwise independence

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(A_i)_{i \in I}$ be a family of events. The family is pairwise independent if $A_i$ and $A_j$ are independent for all $i, j \in I$ with $i \ne j$, that is, $P(A_i \cap A_j) = P(A_i) \, P(A_j)$.

Remarks

Mutual independence implies pairwise independence, but the converse is false. The node pairwise independent events that are not mutually independent gives an example. The same definition applies to random variables: any two of them are independent. Pairwise independence is enough for some results. Examples are the formula for the variance of a sum and the weak law of large numbers with finite variance.

Explanations

L1 human

Pairwise independence checks the events two at a time. It can happen that each pair of events is independent, but a group of three events is not.