Chain rule (probability)

In probability theory, the chain rule (also called the general product rule) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities. The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities. For two events and , the chain rule states that where denotes the conditional probability of given . An Urn A has 1 black ball and 2 white balls and another Urn B has 1 black ball and 3 white balls. Suppose we pick an urn at random and then select a ball from that urn. Let event be choosing the first urn, i.e. , where is the complementary event of . Let event be the chance we choose a white ball. The chance of choosing a white ball, given that we have chosen the first urn, is The intersection then describes choosing the first urn and a white ball from it. The probability can be calculated by the chain rule as follows: For events whose intersection has not probability zero, the chain rule states For , i.e. four events, the chain rule reads We randomly draw 4 cards without replacement from deck of skat with 52 cards. What is the probability that we have picked 4 aces? First, we set . Obviously, we get the following probabilities Applying the chain rule, Let be a probability space. Recall that the conditional probability of an given is defined as Then we have the following theorem. Let be a probability space. Let . Then The formula follows immediately by recursion where we used the definition of the conditional probability in the first step. For two discrete random variables , we use the eventsand in the definition above, and find the joint distribution as or where is the probability distribution of and conditional probability distribution of given . Let be random variables and .