Calculate P(A or B) for mutually exclusive, non-exclusive, and three-event scenarios with stacked probability bars, comparison tables, and repeated-trial analysis.
The OR probability calculator computes P(A or B) — the probability that at least one of two (or three) events occurs. It supports three modes: mutually exclusive events (P(A∪B) = P(A) + P(B)), non-exclusive events (P(A∪B) = P(A) + P(B) − P(A∩B)), and three-event inclusion-exclusion.
Understanding OR probability is essential for risk assessment ("probability of failure A or failure B"), insurance ("probability of claim type A or B"), and everyday probability reasoning. The addition rule is one of the most commonly needed probability formulas.
This calculator visualizes the probability breakdown with stacked bars, compares all probability components side-by-side, and shows how the probability of "at least one" occurrence scales across repeated independent trials. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards.
The addition rule is one of the most frequently needed probability calculations, from simple "what are the odds of X or Y?" questions to complex risk assessments involving multiple failure modes. This calculator prevents double-counting errors and provides visual confirmation that the probabilities partition correctly. Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain.
Mutually exclusive: P(A ∪ B) = P(A) + P(B). Non-exclusive: P(A ∪ B) = P(A) + P(B) − P(A∩B). Three events (inclusion-exclusion): P(A∪B∪C) = ΣP − ΣP(pairs) + P(A∩B∩C).
Result: P(A or B) = 44%
P(A∪B) = 30% + 20% − 6% = 44%. The 6% overlap is subtracted to avoid double-counting events where both A and B occur.
For n events, the inclusion-exclusion formula alternates adding and subtracting intersections: P(A₁∪...∪Aₙ) = ΣP(Aᵢ) − ΣP(Aᵢ∩Aⱼ) + ΣP(Aᵢ∩Aⱼ∩Aₖ) − ... This formula has 2ⁿ−1 terms, making it impractical for many events. In such cases, the complement method is preferred.
System reliability often uses OR probability. A system fails if component A OR component B fails. For independent components with failure probabilities pA and pB, system failure probability = pA + pB − pA×pB. Redundant systems reverse this: the system survives if component A OR component B survives.
De Morgan's laws connect OR and AND through complements: P(A∪B) = 1 − P(A'∩B') and P(A∩B) = 1 − P(A'∪B'). These identities are fundamental for converting between "at least one" and "all/none" probability problems.
OR (union) asks "at least one event occurs" and is P(A) + P(B) − P(A∩B). AND (intersection) asks "both events occur" and is P(A)×P(B) for independent events. OR always gives a larger or equal probability.
When they cannot happen simultaneously. A coin landing heads AND tails is impossible. A single card being both a heart and a spade is impossible. For mutually exclusive events, P(A∩B) = 0.
It divides the total probability space into: only A, both A and B, only B, and neither. These four regions always sum to 100%.
It uses the inclusion-exclusion principle for three events. For independent events, all pairwise and triple intersections are computed from the marginal probabilities.
Each trial has probability (1−P(A∪B)) of "no occurrence." After n trials, P(none) = (1−P(A∪B))^n, which shrinks exponentially. So P(at least one) = 1 − (1−P(A∪B))^n grows rapidly.
No. P(A or B) is capped at 100%. If P(A) + P(B) > 100% and they're "mutually exclusive," the probabilities are inconsistent — genuinely mutually exclusive events can't have P(A) + P(B) > 1.