Law of Addition for Computing Probabilities

The Law of Addition is one of the most basic theorems in Probability. It takes a very clear form when depicting it in a Venn-Diagram: The idea is that when we count probabilities for A or B, when we add \(\Pr(A)\) and \(\Pr(B)\), it happens that we count twice the portion that corresponds to \(\Pr(A \cap B)\).

Hence, the Law of Addition takes the following shape:

\[\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B) \]

Notice that re-arranging the above expression, we land into one version of law of multiplication for probabilities.

\[\Pr(A \cap B) = \Pr(A) + \Pr(B) - \Pr(A \cup B) \]