Probability Union Calculator
Enter the probability of each event and of their overlap to get the probability that at least one of them happens — P(A∪B) = P(A) + P(B) − P(A∩B).
The addition rule, done for you
Enter P(A), P(B), and P(A∩B) and the calculator returns P(A∪B) — the chance that event A or event B (or both) occurs.
Use values 0–1
Every probability must be between 0 and 1, and the overlap P(A∩B) cannot be larger than either single event.
What is the probability of a union?
The chance that A or B happens
The probability union calculator answers a single, common question: what is the chance that at least one of two events happens? In set notation that is P(A∪B), read as "the probability of A union B" or simply P(A or B). The addition rule says you add the two individual probabilities and then subtract their overlap, because anything counted in both A and B would otherwise be counted twice. Enter the probability of event A, the probability of event B, and the probability that both happen together, and the calculator returns the union in the range 0 to 1.
Enter P(A), P(B), and the overlap P(A∩B), each between 0 and 1, to get the probability that A or B happens instantly.
The union of two events is the sum of their probabilities minus the probability they both occur — the overlap is subtracted so it is not counted twice.
P(A∪B) = P(A) + P(B) − P(A∩B)If you simply added P(A) and P(B), the region where both events happen would be included in each term and therefore double-counted. Subtracting P(A∩B) removes that double count exactly once, leaving the true probability of the combined event.
Suppose event A has probability 0.5, event B has probability 0.4, and the chance that both happen is 0.2.
Add the two probabilities
0.5 + 0.4 = 0.9 — but this counts the overlap twice.
Subtract the overlap
0.9 − 0.2 = 0.7 — removing the double-counted intersection.
Read the result
P(A∪B) = 0.7, so there is a 70% chance that A or B (or both) occurs.
The result is the probability that at least one of the two events occurs, expressed as a number between 0 and 1 (multiply by 100 for a percentage). The reason you subtract the overlap is the heart of the addition rule: when both events can happen at the same time, the cases where they both occur sit inside event A and inside event B, so adding the two probabilities counts them twice. Subtracting P(A∩B) corrects for this exactly once. When the two events are mutually exclusive — they cannot both happen — the overlap is zero, P(A∩B) = 0, and the union collapses to the simple sum P(A) + P(B). That is why two outcomes of a single dice roll, which can never both occur, just add up. The larger the overlap between two events, the smaller their union relative to the raw sum, because more of their probability is shared rather than additional.
The addition rule is exact, but the inputs have to be valid probabilities that describe a consistent pair of events.
Inputs must be consistent probabilities
Every input must be a probability between 0 and 1. The overlap P(A∩B) cannot exceed either single event, because an intersection can never be more likely than the events it sits inside, and the resulting union cannot exceed 1. If you enter figures that break these rules the calculator returns no result — check that P(A∩B) is at most the smaller of P(A) and P(B), and that the events are genuinely related to the same experiment.