> [!NOTE] Theorem > Let $(\Omega, \mathcal{F}, \mathbb{R})$ be a [[Probability Space|probability space]]. For all $A, B\in \mathcal{F}$ such that $A$ is a [[Subsets|subset]] of $B,$ the [[Probability Measure|probability measure]] $\mathbb{P}$ satisfies $\mathbb{P}(B-A)=\mathbb{P}(B)-\mathbb{P}(A),$where $B-A$ is a [[Set Difference|set difference]]. **Proof**: We have $B$ is the union of disjoint sets $(B-A)$ and $A.$ Thus by finite additivity, $\mathbb{P}(B)=\mathbb{P}((B-A)\cup A)=\mathbb{P}(B-A)+\mathbb{P}(A)$which proves the claim. **Proof**: Follows from [[Probability of Set Difference of Events]] noting that $A\cap B =A$. # Applications **Consequences**: [[Probability of Complement of Event]]; [[Probability of Empty Set is Zero]]; [[Probability of Subset of Event is Less Than or Equal to Probability of Event]].