TITLE: analog of principle of inclusion-exclusion
QUESTION [16 upvotes]: 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?

REPLY [3 votes]: In belief propagation there is a notion of inclusion-exclusion for computing the join probability distributions of a set of variables, from a set of factors or marginals over subsets of those variables.  For example, suppose {X,Y,Z} is your set of variables, and you know the marginal probabilities for pX,Y(x,y) and pY,Z(y,z).  If these two are compatible, then the marginal pY can be computed in either of the ways
pY(y) = integral pX,Y(x,y) dx = integral pY,Z(y,z) dz
Then a maximum entropy guess at the full joint distrubution is given by inclusion-exclusion over subsets of variables
pX,Y,Z(x,y,z) = pX,Y(x,y) pY,Z(x,y) / pY(y)
You might take a look at "Belief propagation" on wikipedia, or the more techinical article "Constructing Free Energy Approximations and Generalized Belief Propagation Algorithms" by Yedidia, Freeman and Weiss, which uses inclusion-exclusion in the form of 'counting numbers'.