## Definition The ==Inclusion-Exclusion Principle== is a formula for computing the probability (or size) of the union of events by alternately adding and subtracting the probabilities of their intersections to correct for over-counting. For two events: $ P(A \cup B) = P(A) + P(B) - P(A \cap B) $ For three events: $ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) $ > [!example]- In a group of 200 people, 120 speak English, 90 speak French, and 50 speak both. How many speak at least one of the two languages? > By inclusion-exclusion: > $ |E \cup F| = |E| + |F| - |E \cap F| = 120 + 90 - 50 = 160 $ > So 160 people speak at least one language, and $200 - 160 = 40$ speak neither.