> [!NOTE] Definition (Pairwise Independence) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A_{1},A_{2},A_{3},\dots A_{n} \in \mathcal{F}$. The events $A_{1},A_{2},\dots, A_{n}$ are *pairwise independent* iff $A_{j}$ and $A_{k}$ are [[Independence of Two Events|independent]] for all $1 \leq j <k \leq n$. > [!Example] > Two die are rolled. Let $\begin{align} > A_{1} &= \{ \text{the first dice is even} \} \\ > A_{2} &= \{ \text{the second dice is odd} \} \\ > A_{3} &= \{ \text{sum of the die is 7} \} > \end{align}$These events are pairwise independent since $\begin{gathered} > \mathbb{P}(A_1\cap A_2) =\frac14=\mathbb{P}(A_1)\mathbb{P}(A_2) \\ > \mathbb{P}(A_1\cap A_3) =\frac1{12}=\mathbb{P}(A_1)\mathbb{P}(A_3) \\ > \mathbb{P}(A_2\cap A_3) =\frac1{12}=\mathbb{P}(A_2)\mathbb{P}(A_3). > \end{gathered}$However the are [[Mutually Independent Set of Events|mutually dependent]] since $\mathbb{P}(A_1\cap A_2\cap A_3)=\frac1{12}\neq\frac1{24} = \mathbb{P}(A_1)\mathbb{P}(A_2)\mathbb{P}(A_3).$ # Properties ....