Dependence of events, revisited

Independence is a key concept in probability. Conceptually, we think of two events as being independent if the outcome of one event doesn’t affect the outcome of the other and vice versa. Mathematically, we say that events A and B are independent if the probability that both occur is the product of the probabilities that each occurs. More precisely, P ( A ∩ B ) = P ( A ) ( P ( B ) in which P () denotes the probability of the given event. Alternatively, we say that A and B are independent if the conditional probability that A occurs given that B has occurred, P ( A | B ), satisfies P ( A | B ) = P ( A ). That is, whether or not B occurs does not affect whether or not A occurs.