Union & Intersection Probability Calculator

About this calculator

The addition rule of probability states that P(A ∪ B) = P(A) + P(B) − P(A ∩ B): to find the probability that event A or event B (or both) occurs, you add their individual probabilities and then subtract the overlap, because that overlap would otherwise be counted twice. This rule is one of the first non-trivial results taught in any probability or statistics course, and it generalizes the simpler additive rule used for mutually exclusive events (where P(A ∩ B) = 0).

Beyond the classroom, the union-intersection relationship underlies real risk and reliability calculations: an insurance analyst estimating the chance that a claim involves fire damage or water damage needs to subtract the overlap where both occurred; a quality engineer estimating the probability that a product fails test A or test B uses the same correction; marketers analyzing customer segments who bought product A or product B rely on it to avoid double-counting shared customers. Once P(A∪B) is known, the "only A" and "only B" probabilities — P(A) − P(A∩B) and P(B) − P(A∩B) — describe the parts of each event that don't overlap.

This calculator takes P(A), P(B) and P(A∩B) as inputs and instantly returns P(A∪B) along with the only-A and only-B breakdown, so you don't have to work the algebra by hand each time.