1. Independent Events Let A and B be two events. It is quite possible that the percentage of B embodied by A is the same as the percentage of S embodied by A. For example, A could be half of S, and A∩B half of B (A = red die ≥ 4; B = blue die ≤ 3). In formulas, we are saying that

3. In formula we are saying that P(A|B) = P(A) and in words we are saying that knowledge that B happened does not alter the a priori chance of A. In this case we say that A is independent of B From P(A) = P(A|B) = we get P(A). P(B) = P(A∩B)

4. This last formula, because of symmetry, also says that P(B|A) = P(B) Therefore B is independent of A Independence is a symmetric concept ! We give the formal definition (used to verify independence) Definition. Two events A and B are called independent when, and only when, P(A) x P(B) = P(A∩B) Now go do assigned homework.

5. Multiplicative Rule For any two events A and B (independent or not) we have the two equations We get P(A∩B) two ways: Generally, one will be useful, one useless. Both are called Multiplicative Rules.