Inclusion–exclusion principle
Special case

If, in the probabilistic version of the inclusion–exclusion principle, the probability of the intersection A only depends on the cardinality of I, meaning that for every k in {1,..., n } there is an a such that

then the above formula simplifies to

due to the combinatorial interpretation of the binomial coefficient (n k) {\textstyle {\binom {n}{k}}}. For example, if the events A i {\displaystyle A_{i}} are independent and identically distributed, then P (A i) = p {\displaystyle \mathbb {P} (A_{i})=p} for all i, and we have a k = p k {\displaystyle a_{k}=p^{k}}, in which case the expression above simplifies to

(This result can also be derived more simply by considering the intersection of the complements of the events A i {\displaystyle A_{i}}.)

An analogous simplification is possible in the case of a general measure space (S, Σ, μ) and measurable subsets A,..., A of finite measure.
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