can be defined as or symmetrically for odds of given , and thus is 1 if and only if the events are independent. More than two events A finite set of events is pairwise independent if every pair of events is independent—that is, if and only if for all distinct pairs of indices , A finite set of events is mutually independent if every event is independent of any intersection of the other events—that is, if and only if for every and for every k indices , This is called the multiplication rule for independent events. Note that it is not a single condition involving only the product of all the probabilities of all single events; it must hold true for all subsets of events. For more than two events, a mutually independent set of events is (by definition) pairwise independent; but the converse is not necessarily true. Log probability and information content Stated in terms of log probability, two events are independent if and only if the log probability of the joint event is the sum of the log probability of the individual events: In information theory, negative log probability is interpreted as information content, and thus two events are independent if and only if the information content of the combined event equals the sum of information content of the individual events: See for details. For real valued random variables Two random variables Two random variables and are independent if and only if (iff) the elements of the π-system generated by them are independent; that is to say, for every and , the events and are independent events (as defined above in ). That is, and with cumulative distribution functions and , are independent iff the combined random variable has a joint cumulative distribution function or equivalently, if the probability densities and and the joint probability density exist, More than two random variables A finite set of random variables is pairwise independent if and only if every pair of random variables is independent. Even if the set of random variables is pairwise independent, it is not necessarily mutually independent as defined next. A finite set of random variables is mutually independent if and only if for any sequence of numbers , the events are mutually independent events (as defined above in ). This is equivalent to the following condition on the joint cumulative distribution function A finite set of