General Addition Rule

Adding P(A) and P(B) counts every outcome that belongs to both events twice, so the general addition rule refunds one copy of the overlap. Drawing one card from a standard deck, P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 ≈ 0.308: the king of hearts is a single card and gets counted once. This is the two-event case of inclusion–exclusion, the counting principle Abraham de Moivre put to work in The Doctrine of Chances (1718), the book that taught a generation of gamblers and annuity sellers to reason about risk.

Enter probabilities as decimals from 0 to 1 (or use the % unit). Rearranged for the overlap it becomes a quiet diagnostic tool: if a survey reports that 60% of people drink coffee, 50% drink tea, and 80% drink at least one, then P(both) = 0.6 + 0.5 − 0.8 = 0.3, so 30% drink both. If a set of reported figures forces the overlap negative, say 0.3 + 0.3 − 0.9, the numbers are simply inconsistent, and the calculator says so instead of pretending.