Probability Calculator
Compute union, intersection, complement, and exclusive event probabilities for two events.
Event probabilities
P(A and B):
P(A or B):
P(neither A nor B):
P(exactly one):
P(A only):
P(B only):
For independent events, the intersection equals P(A) × P(B). If not independent, provide P(A and B) directly.
Independent vs. dependent events
Two events are independent when one event's outcome doesn't affect the other's probability (like separate coin flips) — in that case, P(A and B) = P(A) × P(B). When events are dependent or related in some way (like drawing cards without replacement), you must supply the actual intersection probability directly rather than relying on that shortcut multiplication.
Frequently asked questions
What does P(A or B) actually include? It's the probability that at least one of the events happens — A alone, B alone, or both together — which is why the formula adds P(A) and P(B) then subtracts the overlap P(A and B) to avoid double-counting.
What's the difference between "exactly one" and "A or B"? "A or B" (union) includes the case where both happen; "exactly one" (exclusive or) specifically excludes the case where both happen — it only counts outcomes where just one of the two events occurs.
Why would P(A and B) not equal P(A) × P(B)? That shortcut only holds for independent events — if the events influence each other (dependent events), their true joint probability can be higher or lower than the simple product, which is why the calculator lets you enter it directly for non-independent cases.