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2.2.1 Joint, Marginal, and Conditional Probabilities     continued...

Any joint probability can be factored into the product of a marginal probability and a conditional probability:

where is the joint probability, is the marginal probability (the probability that event occurs regardless of event ), and is the conditional probability (the probability that event occurs given that event occurs). Note that the marginal probability for event to occur is simply the probability that the event occurs, or . Let us now assume that there are J mutually-exclusive events , and the following identity is evident:

Using Eq. (2), we easily manipulate Eq. (1) to obtain the following expression for the joint probability