> [!NOTE] Theorem (Inclusion-Exclusion Principle) > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $A_{1},A_{2},\dots,A_{n}$ be [[Finite Set|finite sets]]. Then $\mathbb{P} \left(\bigcup_{i=1}^nA_i\right)=\sum_{\emptyset\neq J\subseteq\{1,\ldots,n\}}(-1)^{|J|+1}\mathbb{P}\left(\bigcap_{j\in J}A_j\right).$ **Proof**: Follows from [[Inclusion-Exclusion Principle]] since the [[Probability Measure|probability measure]] $\mathbb{P}$ is, by definition, finitely additive. **Proof by induction**: For $n\geq 2,$ let $P(n)$ be .... By [[Probability Measure is Strongly Additive]], $P(2)$ is true. # Applications **Example**: Using the theorem with $n=3,$ $\mathbb{P}(A \cup B \cup C)=\mathbb{P}(A)+ \mathbb{P}(B)+ \mathbb{P}(C) - \mathbb{P}(A \cap B)- \mathbb{P}(A \cap C)-\mathbb{P}(B \cap C)+ \mathbb{P}(A \cap B \cap C).$