Probability of A or B Calculator
The probability that at least one of two events happens — with the inclusion–exclusion correction that catches the overlap.
P(A or B) = P(A) + P(B) − P(A and B).
How the probability of a or b calculator works
P(A or B) = P(A) + P(B) − P(A and B). The subtraction matters: adding the two probabilities counts the overlap twice. Only when the events are mutually exclusive is the overlap zero and plain addition correct.
If you know the events are independent rather than exclusive, the overlap is P(A) × P(B), which the calculator shows alongside so you can see the difference.
Formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Worked examples
| Inputs | P(A or B) | Note |
|---|---|---|
| Independent 40% and 30% | 58% | 58% |
| Mutually exclusive | 70% | 70% |
| Known overlap of 12% | 58% | 58% |
FAQFrequently asked questions
What is P(A or B)?
The chance that at least one happens. "Or" in probability is inclusive — both counts as a success.
Why subtract the overlap?
Because adding P(A) and P(B) counts the cases where both happen twice. Inclusion–exclusion removes the double count.
When can I just add them?
Only when the events cannot both happen — mutually exclusive. Otherwise the sum overstates the answer.
What if the events are independent?
The overlap is P(A) × P(B), so the union is P(A) + P(B) − P(A)P(B).
Are independent and mutually exclusive the same?
No — they are nearly opposites. Exclusive events are strongly dependent: knowing one happened tells you the other did not.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.