Probability of Three Events Calculator
Enter individual probabilities and overlaps for three events to compute the probability of at least one event happening and the exclusive region probabilities.
P(A ∪ B ∪ C)
75.00%
Probability at least one event occurs
Exclusive A only
18.00%
P(A) minus overlaps
Exclusive B only
15.00%
P(B) minus overlaps
Exclusive C only
15.00%
P(C) minus overlaps
| Region | Probability |
|---|---|
| A only | 18.00% |
| B only | 15.00% |
| C only | 15.00% |
| A ∩ B only | 12.00% |
| A ∩ C only | 7.00% |
| B ∩ C only | 5.00% |
| A ∩ B ∩ C | 3.00% |
| None of the events | 25.00% |
How to Use This Calculator
- Enter the individual probabilities for events A, B, and C.
- Provide the pairwise intersection probabilities and the triple intersection (if known).
- Review the inclusion–exclusion result P(A ∪ B ∪ C) and the exclusive region breakdown.
- Use warnings to adjust inputs if probabilities become inconsistent or exceed logical bounds.
Formula
This is the inclusion–exclusion principle for three events. Subtract pairwise overlaps to avoid double counting, then add back the triple overlap because it was subtracted three times.
Full Description
Probability problems with multiple overlapping events frequently arise in quality control, epidemiology, and risk analysis. Inclusion–exclusion offers a systematic way to combine overlapping probabilities without double counting. Breaking results into exclusive regions helps visualize Venn diagrams and answer questions such as “What portion of the population experiences exactly two of the events?”
When intersection data is missing, consider estimating overlaps based on domain knowledge or assuming independence (P(A ∩ B) = P(A)P(B)), though real-world events rarely behave independently.
Frequently Asked Questions
What if I don’t know the intersections?
Without intersection data, the union probability cannot be determined uniquely. You may bound the result using Boole’s inequality or assume independence as an approximation.
Why do I get negative region probabilities?
Negative values indicate inconsistent inputs — for example, specifying overlaps larger than individual event probabilities. Adjust the numbers until warnings disappear.
Can probabilities exceed 1?
No. The calculator clamps values between 0 and 1 and flags if inclusion–exclusion yields a union exceeding 1, signaling inconsistent inputs.
How does this help with Venn diagrams?
The exclusive region breakdown directly maps to the seven regions of a three-set Venn diagram, enabling clear visualization and further probability calculations.