> [!NOTE] Theorem > Let $(\Omega, \mathcal{F}, \mathbb{R})$ be a [[Probability Space|probability space]]. For all $A, B\in \mathcal{F},$ the [[Probability Measure|probability measure]] $\mathbb{P}$ satisfies $\mathbb{P}(B-A)=\mathbb{P}(B)-\mathbb{P}(A\cap B),$where $B-A$ is a [[Set Difference|set difference]]. **Proof**: By [[Set Difference and Intersection From Partition]], $B$ is the union of the two disjoint sets $B-A$ and $A \cap B.$ Thus by finite additivity, $\mathbb{P}(B-A)=\mathbb{P}((B-A)\cup (A\cap B))=\mathbb{P}(B-A)+\mathbb{P}(A \cap B).$ # Application **Corollary**: See [[Probability of Subset of Event]].