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 n-part experiment with n 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.