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