Probability of Union Calculator
The probability that at least one of two events happens is one of the first things you learn in any statistics course, and it trips people up because of a subtle double count. This calculator applies the addition rule correctly so you do not have to worry about it. Enter the probability of event A, the probability of event B, and the probability that both happen together, P(A and B), and the tool adds the first two and subtracts the overlap to give P(A or B), the probability of the union. The reason for subtracting the overlap is simple: when you add P(A) and P(B), any outcome that belongs to both events is counted twice, so subtracting P(A and B) brings each outcome back to a single count. If the events are mutually exclusive, set the overlap to zero and the rule reduces to plain addition. All three inputs are probabilities between 0 and 1, and a consistent set always gives a union between 0 and 1. Use it for dice and card problems, risk analysis or any time you need the chance of A or B. Every figure is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator's defaults.
The probability of a union uses the addition rule: P(A or B) = P(A) + P(B) - P(A and B). With P(A) = 0.50, P(B) = 0.40 and P(A and B) = 0.20, the union is 0.70.
Addition rule formula
P(A or B) = P(A) + P(B) - P(A and B)
P(A) = probability event A occurs
P(B) = probability event B occurs
P(A and B) = probability both occur (0 if mutually exclusive)
Add the two individual probabilities, then subtract the overlap so shared outcomes are counted once. The result is the probability that at least one event occurs.
Worked example
Suppose P(A) = 0.5, P(B) = 0.4 and the events overlap with P(A and B) = 0.2.
- P(A) + P(B) = 0.5 + 0.4 = 0.9
- Subtract the overlap: 0.9 - 0.2 = 0.7
- P(A or B) = 0.70
The probability of the union is 0.70. These are the calculator's default inputs, so the result above matches the widget exactly.
Union for sample inputs
P(A or B) for a few sets of inputs.
| P(A) | P(B) | P(A and B) | P(A or B) |
|---|---|---|---|
| 0.50 | 0.40 | 0.20 | 0.70 |
| 0.30 | 0.30 | 0.00 | 0.60 |
| 0.60 | 0.50 | 0.30 | 0.80 |
| 0.25 | 0.25 | 0.0625 | 0.4375 |
The last row uses independent events, where P(A and B) = P(A) x P(B).
Probability of union calculator: frequently asked questions
What is the probability of a union?
The union of two events A and B is the event that at least one of them occurs, written A or B. Its probability is P(A or B) = P(A) plus P(B) minus P(A and B). You add the two individual probabilities, then subtract the overlap so the shared outcomes are not counted twice. This is the general addition rule of probability.
Why subtract P(A and B)?
When you add P(A) and P(B), any outcomes that belong to both events get counted twice, once in each term. Subtracting P(A and B), the probability of the overlap, removes that double count so each outcome is counted exactly once. If the events cannot happen together, the overlap is zero and there is nothing to subtract.
What if the events are mutually exclusive?
Mutually exclusive events cannot both occur, so their overlap P(A and B) is zero. The addition rule then simplifies to P(A or B) = P(A) plus P(B). Enter zero for P(A and B) in that case. For example, rolling a 2 or a 5 on one die are mutually exclusive, so you simply add their probabilities.
Can the result be more than 1?
No, a valid probability is always between 0 and 1. If your inputs are consistent, the union cannot exceed 1. If you get a result above 1, your inputs are inconsistent, usually because P(A and B) is too small for the P(A) and P(B) you entered. Check that the overlap is at least P(A) plus P(B) minus 1.
How do I find P(A and B)?
If the events are independent, P(A and B) equals P(A) times P(B). If they are not independent, you need the joint probability from the problem, a table or data. For mutually exclusive events it is zero. Enter whichever joint probability applies; this calculator uses it directly in the addition rule.
Official sources
- Probability and statistics reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.