Probability

The probability of event E given

The probability of event E given that event F has already occurred is P(E|F)=P(E∩F)/P(F), provided P(F)≠0. It means the sample space is restricted to

For any event F with P(F)≠0

For any event F with P(F)≠0, P(S|F)=1, P(E'|F)=1-P(E|F), and P((A∪B)|F)=P(A|F)+P(B|F)-P((A∩B)|F). If A and B are disjoint, then P((A∪B)|F)=P(A|F)+P(B|

For two events

For two events, P(E∩F)=P(E)·P(F|E)=P(F)·P(E|F), provided P(E)≠0 and P(F)≠0. For three events, P(E∩F∩G)=P(E)P(F|E)P(G|EF).

Two events E and F

Two events E and F are independent if P(F|E)=P(F) and P(E|F)=P(E), and equivalently if P(E∩F)=P(E)·P(F). Independent events with nonzero probability c

Three events

Three events A, B, and C are mutually independent if all pairwise products and the triple product hold: P(A∩B)=P(A)P(B), P(A∩C)=P(A)P(C), P(B∩C)=P(B)P