We now consider an experiment that consists of two parts, and each part leads
to the occurrence of specified events. Let us define events arising from the
first part of the experiment by 
with probability 
and events from
the second part by 
with probability 
.
The combination of events and may be called a
composite event,
denoted by the ordered pair 
.
We wish to generalize the definition of probability to apply to the composite
event 
.
The joint probability 
is defined to be the probability that
the first
part of the experiment led to event and the second part of the experiment
led to event .
Thus, the joint probability is the probability that the composite
event occurred (i.e., the probability that both events and
occur).
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 
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
Using Eq. (1), Eq. (3) leads to the following expression for the conditional probability:
It is important to note that the joint probability , the marginal
probability
, and the conditional probability are all legitimate
probabilities,
hence they satisfy the properties given in the box above.
Finally, it is straightforward to generalize these definitions to treat a
three-part experiment that
has a composite event consisting of three events, or in general an 
-part
experiment with events occurring.
If events and are independent, then the probability of one
occurring does not affect the probability of the other occurring, therefore:
Using Eq. (4), Eq. (5) leads immediately to
for independent events and .
This last equation reflects the fact that the
probability of event occurring is independent of whether event has
occurred, if events and are independent.
