Independence of events A and B is formally defined as P(A∩B) = P(A)P(B), generalized to n events by requiring every subset's joint probability to equal the product of individual probabilities (a strictly stronger condition than pairwise independence alone); this is distinct from disjointness, which describes mutually exclusive events. Conditional probability, P(A|B) = P(A∩B)/P(B), formalizes how a probability should be updated once it is learned that another event has occurred, and is justified via two equivalent interpretations: restricting and renormalizing a discrete probability-weighted sample space to the subset where the conditioning event holds, and the long-run frequency of an event among repeated trials in which the conditioning event occurred. From this definition follow the multiplication rule, its generalization to sequential conditioning of many events (valid under any ordering), and Bayes' rule (relating P(A|B) to P(B|A)), all foundational to probability theory's treatment of updating belief under evidence.
Independence of events A and B is formally defined as P(A∩B) = P(A)P(B), generalized to n events by requiring every subset's joint probability to equal the product of individual probabilities (a stri…