Pairwise Independent Events

Why

Is a sufficient condition for mutual independence of a set of events the fact that each pair of events of the set are independent? The surprising answer is no.

Definition

Suppose $P$ is a event probability function on a finite sample space $\Omega $. The events $A_1, \dots , A_n$ are independent (or mutually independent), if for all indices $i \neq j$.

\[ P(A_{i} \cap A_{j}) = P(A_{i}) P(A_{j}) \]

Clearly, if $A_1, \dots , A_n$ are mutually independent, then they are pairwise independent. The converse, however, is not true.

Counterexample: two tosses

As usual, model tossing a coin twice with the sample space $\Omega = \set{0,1}^2$. Put a distribution $p: \Omega \to [0,1]$ so that $p(\omega ) = 1/4$ for all $\omega \in \Omega $. Define $A = \set{(1,0),(1,1)}$ (the first toss is heads) , $B = \set{(0,1),(1,1)}$ (the second toss is heads), and $C = \set{(0,0),(1,1)}$. Then $P(A) = P(B) = P(C) = 1/2$, and

\[ P(A \cap B) = P(B \cap C) = P(A \cap C) = 1/4. \]

Hence $A, B, C$ are pairwise independent. However,

\[ P(A \cap B \cap C) = 1/4 \neq P(A)P(B)P(C). \]

So $A, B, C$ are not mutually independent.