When I teach elementary probability to my finite math students, a common error is to mix up the concepts of disjointness and independence. At some point I thought that it might be helpful to some students to draw the analogy between the two concepts implied by the following pair of statements:

• To compute the probability of the union of disjoint events, you add the probabilities of the events.

• To compute the probability of the intersection of independent events, you multiply the probabilities of the events.

I also teach them is that when events are not disjoint, you can still compute the probability of their union by applying the principle of inclusion-exclusion. Hence the question: Is there a useful analog of the principle of inclusion-exclusion for computing the probability of the intersection of non-independent events?

Edit: I am incorporating the following clarification that I made in a comment responding to the answer of Anna Varvak:

In inclusion-exclusion, one alternately adds and subtracts intersections. Intersections measure the degree to which disjointness fails. Can we write the right-hand side of Bayes Theorem as alternate multiplications and divisions of something, where "something" measures the degree to which independence fails?

When I teach elementary probability to my finite math students, a common error is to mix up the concepts of disjointness and independence. At some point I thought that it might be helpful to some students to draw the analogy between the two concepts implied by the following pair of statements:

• To compute the probability of the union of disjoint events, you add the probabilities of the events.

• To compute the probability of the intersection of independent events, you multiply the probabilities of the events.

I also teach them is that when events are not disjoint, you can still compute the probability of their union by applying the principle of inclusion-exclusion. Hence the question: Is there a useful analog of the principle of inclusion-exclusion for computing the probability of the intersection of non-independent events?