Three overlapping transparent colored disks beside eight small compartments. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Build the union from nonoverlapping regions
For three events, adding a+b+c counts every two-event overlap twice and the triple overlap three times. Subtract ab, ac and bc, then add abc once: P(A or B or C) = a+b+c−ab−ac−bc+abc. The final triple adjustment is necessary because the three pairwise subtractions otherwise remove that region completely.
Check a mutually independent example
Let a = b = c = 50% and assume mutual independence. Each pairwise intersection is 25% and the triple intersection is 12.5%. The union is 87.5%. Exactly one event has probability 37.5%, exactly two has probability 37.5%, all three 12.5% and none 12.5%. These four disjoint categories sum to 100%.
Validate all eight atoms
The atoms are ABC, AB only, AC only, BC only, A only, B only, C only and none. For example, A-only equals a−ab−ac+abc. Every atom must be nonnegative. Three pairwise intersections can individually satisfy two-event bounds and still be jointly incompatible with the supplied triple intersection.
Use the full model
The Probability of 3 Events Calculator reports the atoms and validates them before reporting the union. It offers mutual independence as an explicit assumption, not something inferred from separate marginal values. Read pairwise versus mutual independence for a counterexample. For only two events, the two-event Probability Calculator is the simpler workspace.
Check the model behind your result
Use the Probability of 3 Events Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26