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To compute the probability that exactly k of n events occur, use generalized inclusion-exclusion

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Quick description

Suppose that , , ..., are events and you want to calculate the probability that exactly of these events occur. That is, you would like to compute the probability of the set of samples that belong to exactly of these sets. One can use the following generalized inclusion-exclusion formula to accomplish this:

The formula looks complicated, but if the intersection probabilities are simple to compute it gives a simple method to compute the desired probability. It can be useful in problems where it is not easy to use recursion or other techniques and when has a complex dependence structure. In its limitations and usefulness it is similar to the ordinary inclusion exclusion principle.

Prerequisites

Elementary probability.