Principle of Inclusion-Exclusion for Probability

Based on the Principle of Inclusion-Exclusion, which relates to Cardinality. We can simply divide both sides by the cardinality of the Universe in order to get everything in terms of cardinality. See Correspondance to Set Theory

$$\begin{array}{rl}P(A_1\cup A_2\cup \cdots \cup A_n)& =\\ \sum _{i=1}^{n}P(A_i)& -\\ \sum _{1\le i<j\le n}P(A_i\cap A_j)& +\\ \sum _{1\le i<j<k\le n}P(A_i\cap A_j\cap A_k)& -\\ \dots & \\ & \\ +(-1{)}^{n+1}P(A_1\cap A_2\cap \cdots \cap A_n)& \end{array}$$