Inclusion-Exclusion Rule

Rule

Two Events

• For any two events $A$, $B$
• $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Generalized Events

• For $n$ events $A_1,\dots ,A_n$
• $P({\cup }_{i=1}^n{A}_i)={\sum }_{i=1}^{n}P(A_i)-{\sum }_{j=1}^{i}P(A_i\cap A_j)+{\sum }_{k=1}^{j}P(A_1\cap A_j\cap A_k)$
• $⟹P({\cup }_{i=1}^n{A}_i)=(-1{)}^{n-1}P(A_1\cap \cdots \cap A_n)$