Combine two events — AND, OR, conditional, complements, and "at least one" — with a live Venn diagram.
P(A and B) — the intersection — is the probability that both events happen.
P(A or B) — the union — is the probability that at least one happens, and equals P(A) + P(B) − P(A and B). You subtract the overlap so it isn't double-counted.
Independent: one event happening doesn't change the other's probability, so P(A and B) = P(A)·P(B). Example: two separate coin flips.
Mutually exclusive: the events cannot both happen, so P(A and B) = 0. Example: a single die showing both a 3 and a 5. These are not the same thing — in fact mutually exclusive events are strongly dependent (if one happens, the other definitely didn't).
P(A|B) is the probability of A given that B has occurred. It's calculated as P(A and B) / P(B) — you restrict attention to the world where B happened. If A and B are independent, P(A|B) = P(A), because knowing B tells you nothing about A.
For two events, P(at least one) = P(A or B) = P(A) + P(B) − P(A and B). For many independent events, the fastest route is the complement rule: P(at least one) = 1 − P(none happen). For example, the chance of at least one head in 3 fair coin flips is 1 − (½)³ = 0.875.
Probabilities have to be internally consistent. In general mode, P(A and B) can't exceed either P(A) or P(B), and P(A or B) can't exceed 1. If your numbers break these rules, no real pair of events could produce them, so the calculator flags it.