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

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Quick description

Suppose that $A_1$ , $A_2$ , ..., $A_n$ are events and you want to calculate the probability that exactly $k$ of these events occur. That is, you would like to compute the probability $p_{[m]}$ of the set of samples that belong to exactly $k$ of these sets. One can use the following generalized inclusion-exclusion formula to accomplish this:

$p_{[m]}= \sum_{k=0}^{N-m} (-1)^{k} \binom{ m+ k}{m} \sum_{1\le i_1 < i_2 < \cdots < i_{m+k} \le n } P\left( \cap_{j=1}^{m+k} A_{i_j}\right)$

The formula looks complicated, but if the intersection probabilities $P\left( \cap_{j=1}^k A_{i_j}\right)$ 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 $P$ has a complex dependence structure. In its limitations and usefulness it is similar to the ordinary inclusion exclusion principle.

Prerequisites

Elementary probability.

Example 1

General discussion

See the article on inclusion-exclusion principle.

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