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General Addition Rule

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Worked example: Coffee 0.6, tea 0.5, either 0.8 → both = 0.3 — press Try an example to run it live, then adjust anything.

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General Addition Rule explained

P(A)P(B)P(A∩B)

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.

General Addition Rule formula

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
Where
  • P(A∪B)P(A \cup B)= Probability of A or B
  • P(A)P(A)= Probability of A
  • P(B)P(B)= Probability of B
  • P(A∩B)P(A \cap B)= Probability of A and B

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