of , given "; the conditioning applies to the whole statement: "( is independent of ) given ". This notation extends for " is independent of ." If assumes a countable set of values, this is equivalent to the conditional independence of X and Y for the events of the form . Conditional independence of more than two events, or of more than two random variables, is defined analogously. The following two examples show that neither implies nor is implied by . First, suppose is 0 with probability 0.5 and 1 otherwise. When W = 0 take and to be independent, each having the value 0 with probability 0.99 and the value 1 otherwise. When , and are again independent, but this time they take the value 1 with probability 0.99. Then . But and are dependent, because Pr(X = 0) < Pr(X = 0|Y = 0). This is because Pr(X = 0) = 0.5, but if Y = 0 then it's very likely that W = 0 and thus that X = 0 as well, so Pr(X = 0|Y = 0) > 0.5. For the second example, suppose , each taking the values 0 and 1 with probability 0.5. Let be the product . Then when , Pr(X = 0) = 2/3, but Pr(X = 0|Y = 0) = 1/2, so is false. This is also an example of Explaining Away. See Kevin Murphy's tutorial where and take the values "brainy" and "sporty". Conditional independence of random vectors Two random vectors and are conditionally independent given a third random vector if and only if they are independent in their conditional cumulative distribution given . Formally: where , and and the conditional cumulative distributions are defined as follows. Uses in Bayesian inference Let p be the proportion of voters who will vote "yes" in an upcoming referendum. In taking an opinion poll, one chooses n voters randomly from the population. For i = 1, ..., n, let Xi = 1 or 0 corresponding, respectively, to whether or not the ith chosen voter will or will not vote "yes". In a frequentist approach to statistical inference one would not attribute any probability distribution to p (unless the probabilities could be somehow interpreted as relative frequencies of occurrence of some event or as proportions of some population) and one would say that X1, ..., Xn are independent random variables. By contrast, in