Calculate combined probability of three events using inclusion-exclusion. Supports independent and dependent events with Venn diagram regions, conditional probabilities, and repeated trials.
The probability of three events calculator works through the full three-set picture for events A, B, and C. It calculates the union P(A∪B∪C), the exactly-k breakdown, all eight Venn regions, and the conditional probabilities that fall out of the same data.
Use the independent mode when pairwise and triple intersections should be derived from the marginal probabilities. Use the dependent mode when intersections are known from data or when the events influence one another, such as related test results, overlapping risks, or linked production defects.
The result is a compact view of a problem that is easy to miscount by hand because every overlap changes the final total.
Three-event probability problems are awkward to do manually because every overlap changes the answer. This calculator keeps the inclusion-exclusion bookkeeping in one place, which is useful when you need the union, the exactly-one / exactly-two / exactly-three split, or the probability of no events occurring at all.
It is most useful when the same three outcomes can happen together in different combinations, such as independent component failures, overlapping survey responses, or multi-stage decision flows.
P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C). P(exactly 1) = P(A) + P(B) + P(C) − 2P(A∩B) − 2P(A∩C) − 2P(B∩C) + 3P(A∩B∩C).
Result: P(A∪B∪C) = 68.5%, P(A∩B∩C) = 3%, P(none) = 31.5%
With independent events, P(A∩B) = 12%, P(A∩C) = 10%, P(B∩C) = 7.5%, P(A∩B∩C) = 3%. Applying inclusion-exclusion: 40 + 30 + 25 − 12 − 10 − 7.5 + 3 = 68.5%. Exactly one event occurs 47.5% of the time.
For two events, the union only needs one subtraction term. For three events, each pairwise overlap must be removed once and the triple overlap added back once:
P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C).
That alternating pattern is the core reason three-event problems are harder than they first appear.
In the independent case, the calculator derives every intersection from the marginals. In the dependent case, the intersections must already be consistent with one another. That makes the calculator useful for checking whether a set of claimed probabilities is mathematically possible.
The eight Venn regions correspond to every possible on/off combination of A, B, and C. The table is the easiest way to spot whether the mass is concentrated in shared overlaps, mostly in single-event regions, or spread evenly across the sample space.
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It prevents double-counting when computing the union of events. For three events: add individual probabilities, subtract pairwise intersections, then add back the triple intersection. Without this correction, overlapping regions would be counted multiple times.
Events are independent if P(A∩B) = P(A)×P(B) for every pair. In practice, use domain knowledge: coin flips are independent, but exam scores in related subjects are not. The pairwise comparison table helps check independence numerically.
The inclusion-exclusion principle extends to any number of events, but the formula grows exponentially (2^n − 1 terms). For four events, you'd need 15 terms. This calculator is designed specifically for three events.
It means one and only one of A, B, C occurs — the other two do not. This is the exclusive-or (XOR) for three events. It equals P(A only) + P(B only) + P(C only).
Negative values mean the input probabilities are inconsistent — the intersections violate probability axioms. For example, P(A∩B) cannot exceed P(A) or P(B). Adjust your dependent-event inputs to ensure mathematical consistency.
Each trial independently draws the three events. The chance of "all three" happening at least once in n trials is 1 − (1 − P(A∩B∩C))^n. This assumes trials are independent of each other.