What’s the difference between P(A|B) and P(A∩B)?+−
P(A∩B) is the probability that BOTH A and B happen, out of everything that could happen. P(A|B) is the probability A happens GIVEN that B has already happened — it's calculated only within the subset of outcomes where B occurred, which is why P(A|B) = P(A∩B)/P(B), not just P(A∩B) on its own.
What does it mean if two events are independent?+−
Events A and B are independent if knowing one happened tells you nothing about whether the other happened — mathematically, P(A∩B) = P(A) × P(B). This calculator checks this automatically and tells you whether your entered values are consistent with independence.
Can two events be both mutually exclusive and independent?+−
Only if one of them has probability zero. In general, no — mutually exclusive events (P(A∩B) = 0) are actually a strong form of DEPENDENCE: knowing A happened tells you B definitely did not happen. True independence and mutual exclusivity are opposite relationships, not variations of the same idea.
Why do I need P(A∩B) as an input — can’t it be calculated from P(A) and P(B) alone?+−
No — P(A) and P(B) alone don't determine how much A and B overlap. Two events with the same individual probabilities can have very different amounts of overlap depending on the relationship between them, so the overlap (P(A∩B)) must be given or measured directly, not derived from the individual probabilities.