first trial and that of drawing a red card on the second trial are not independent, because a deck that has had a red card removed has proportionately fewer red cards. Pairwise and mutual independence Consider the two probability spaces shown. In both cases, and . The random variables in the first space are pairwise independent because , , and ; but the three random variables are not mutually independent. The random variables in the second space are both pairwise independent and mutually independent. To illustrate the difference, consider conditioning on two events. In the pairwise independent case, although any one event is independent of each of the other two individually, it is not independent of the intersection of the other two: In the mutually independent case, however, Triple-independence but no pairwise-independence It is possible to create a three-event example in which and yet no two of the three events are pairwise independent (and hence the set of events are not mutually independent). This example shows that mutual independence involves requirements on the products of probabilities of all combinations of events, not just the single events as in this example. Conditional independence For events The events and are conditionally independent given an event when . For random variables Intuitively, two random variables and are conditionally independent given if, once is known, the value of does not add any additional information about . For instance, two measurements and of the same underlying quantity are not independent, but they are conditionally independent given (unless the errors in the two measurements are somehow connected). The formal definition of conditional independence is based on the idea of conditional distributions. If , , and are discrete random variables, then we define and to be conditionally independent given if for all , and such that . On the other hand, if the random variables are continuous and have a joint probability density function , then and are conditionally independent given if for all real numbers , and such that . If discrete and are conditionally independent given , then for any , and with . That is, the conditional distribution for given and is the same as that given alone. A similar equation holds for the conditional probability density functions in the continuous case. Independence can be seen as a special kind of conditional independence, since probability can be seen as a kind of conditional probability