On this page
Read at least one as an inclusive union
The fictional model supplies two event probabilities and their overlap. “At least one” includes cases in which both events occur. It differs from “exactly one,” which excludes the overlap.
Reconstruct the four disjoint cells
A only is 45% − 15% = 30%. B only is 30% − 15% = 15%. Both is 15%. Neither is the remaining 40%. All four cells are nonnegative and sum to one.
| Cell | Probability |
|---|---|
| A only | 30% |
| B only | 15% |
| Both | 15% |
| Neither | 40% |
| Total | 100% |
Add the cells belonging to the union
At least one has probability 30% + 15% + 15% = 60%. Equivalently, 45% + 30% − 15% = 60%. Adding the marginals without subtracting the overlap counts the both-events cell twice.
The overlap is included once in the union.
Distinguish exactly one
Exactly one has probability 30% + 15% = 45%. It is a different event from at least one. Writing out the cells makes the distinction explicit before applying a formula.