How this calculator works
Inclusion-exclusion first adds the three marginal probabilities, subtracts all pairwise intersections, then restores the triple intersection. Pairwise overlap inputs include the triple region. The calculator also resolves the model into all eight exclusive outcomes and checks that none has negative probability, allowing only 1e-12 of arithmetic drift. Marginals alone cannot determine dependent overlaps. Independence here is mutual independence; pairwise independence alone does not determine the triple intersection.
Formula
P(A or B or C) = a + b + c - ab - ac - bc + abc. P(exactly two) = ab + ac + bc - 3abc. P(exactly one) = a + b + c - 2(ab + ac + bc) + 3abc.
Worked example
Three independent events with probability 1/2 each produce eight equally likely outcomes. All three has probability 1/8, at least one has 7/8, exactly one has 3/8 and exactly two has 3/8.
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Frequently asked questions
Do pairwise overlaps include the third event?
Yes. P(A and B) includes outcomes where C also occurs. The table separately shows A and B only.
Why can valid-looking percentages be rejected?
The percentages must fit one joint distribution. They can each lie between zero and one while still implying a negative exclusive region.
Is pairwise independence enough for the default mode?
No. Three events can be pairwise independent while the triple overlap differs from the product. Default mode assumes mutual independence.