> [!NOTE] Theorem > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A,B\in \mathcal{F}$ such that $A\cap B = \emptyset$ (that is, $A$ and $B$ are [[Mutually Exclusive (Disjoint) Events|mutually exclusive]]). The events $A$ and $B$ are [[Independence of Two Events|independent]] iff $\mathbb{P}(A )> 0$ or $\mathbb{P}(B)>0.$ **Proof**: By [[Probability of Empty Set is Zero]], $\mathbb{P}(A\cap B) = \mathbb{P}(\emptyset)=0$thus $\mathbb{P}(A)\cdot \mathbb{P}(B)=0$which gives $\mathbb{P}(A)=0$ or $\mathbb{P}(B)=0.$