OR Probability Calculator
Enter the probability of two events to find the probability that at least one occurs (inclusive OR), exactly one occurs (exclusive OR/XOR), both occur, or neither occurs. Choose whether the events are mutually exclusive or independent, and see the full formula worked out with your numbers.
What does OR probability mean?
In probability, the word 'or' means at least one of the events occurs. Asking 'what is the probability of A or B?' is asking for the probability of the union of the two events, written P(A union B) or P(A or B). This includes three sub-scenarios: A occurs but B does not, B occurs but A does not, and both A and B occur together. The formula depends on whether the events can both happen at the same time.
Inclusive OR vs exclusive OR
Everyday language is often ambiguous. In mathematics and logic, 'inclusive OR' (written A or B) means at least one of A or B is true, and it includes the case where both are true. 'Exclusive OR' (XOR) means exactly one is true, excluding the case where both occur simultaneously. This calculator gives you both values. If you are calculating the probability that you win prize A or prize B in a lottery where winning both is possible, you want inclusive OR. If the contest rules say you can only claim one prize, you want XOR.
Independent vs mutually exclusive events
Two events are mutually exclusive if they cannot both occur in the same trial. Rolling a 2 and rolling a 5 on the same single die toss are mutually exclusive: only one result comes up. For mutually exclusive events, P(A and B) = 0, so the addition rule simplifies to P(A or B) = P(A) + P(B). Two events are independent if the outcome of one has no effect on the other. Flipping a coin and drawing a card from a deck are independent. For independent events, P(A and B) = P(A) x P(B), so the full inclusion-exclusion rule applies: P(A or B) = P(A) + P(B) - P(A) x P(B). Note that mutually exclusive events with non-zero probability are actually dependent, not independent, because knowing A occurred tells you B cannot have occurred.
Worked example: rolling a die
Suppose you roll a fair six-sided die once and want to know the probability of rolling a 1 or a 6. These two outcomes are mutually exclusive (the die can only show one face). P(1) = 1/6 and P(6) = 1/6. Using the mutually exclusive formula: P(1 or 6) = 1/6 + 1/6 = 2/6 = 33.33%. Now suppose you want the probability of rolling an even number or a number greater than 4. The even numbers are 2, 4, 6 and the numbers greater than 4 are 5 and 6, so these overlap at 6. P(even) = 3/6 = 0.5, P(greater than 4) = 2/6 = 0.333, P(both: rolling a 6) = 1/6 = 0.167. Using the inclusion-exclusion formula: P(even or greater than 4) = 0.5 + 0.333 - 0.167 = 0.667 = 66.67%.
OR probability formulas by event type
| Scenario | Formula | Note |
|---|---|---|
| Independent events (non-mutually exclusive) | P(A or B) = P(A) + P(B) - P(A)P(B) | Subtract the overlap to avoid double-counting |
| Mutually exclusive events | P(A or B) = P(A) + P(B) | No overlap possible: P(A and B) = 0 |
| Exactly one (XOR) | P(A XOR B) = P(A) + P(B) - 2P(A)P(B) | Removes the both-occur scenario from the union |
| Neither event | P(neither) = 1 - P(A or B) | Complement of the union |
| Complement of A | P(A') = 1 - P(A) | Probability A does not occur |
Use the correct formula based on whether your events can both occur at the same time.
Frequently asked questions
What is the formula for P(A or B)?
For independent (non-mutually exclusive) events: P(A or B) = P(A) + P(B) - P(A) x P(B). For mutually exclusive events (which cannot both occur): P(A or B) = P(A) + P(B). The second formula is a simplification of the first when P(A and B) = 0.
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot occur at the same time. If one happens, the other is impossible in the same trial. Independent events can occur at the same time, and the occurrence of one does not change the probability of the other. A common mistake is to treat mutually exclusive events as independent: they are actually dependent, because knowing one occurred tells you the other did not.
Why do we subtract P(A and B) in the OR formula?
When you add P(A) and P(B), you count any outcome where both events occur twice: once when you count A and once when you count B. Subtracting P(A and B) removes that double-count so each outcome is counted exactly once. This is called the inclusion-exclusion principle.
What is exclusive OR (XOR) probability?
XOR probability is the probability that exactly one of the two events occurs, but not both. The formula is P(A XOR B) = P(A or B) - P(A and B), which equals P(A) + P(B) - 2 x P(A) x P(B) for independent events. XOR is useful when outcomes that satisfy both conditions should not be counted, for example a promotion that goes to exactly one candidate.
How do I calculate 'neither A nor B'?
P(neither) = 1 - P(A or B). This is the complement of the union. For example, if there is a 70% chance that event A or event B occurs, there is a 30% chance that neither occurs.
Can P(A or B) be greater than 1?
No. Probabilities are always between 0 and 1 (or 0% and 100%). If you apply P(A) + P(B) without subtracting the overlap for non-mutually exclusive events, the sum can exceed 1, which is why the full inclusion-exclusion formula is required. For mutually exclusive events, P(A) + P(B) must also not exceed 1, because P(A or B) cannot be greater than 1.