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TL;DR

For the union of sets: \( |A_1 \cup \cdots \cup A_n| = \sum|A_i| - \sum|A_i \cap A_j| + \cdots + (-1)^{n+1}|A_1 \cap \cdots \cap A_n| \). Generalizes to probabilities by replacing cardinalities with probabilities.

By Valenke Exam Prep Team·Last updated 2026-06-03

Inclusion-Exclusion Principle

For the union of sets: \( |A_1 \cup \cdots \cup A_n| = \sum|A_i| - \sum|A_i \cap A_j| + \cdots + (-1)^{n+1}|A_1 \cap \cdots \cap A_n| \). Generalizes to probabilities by replacing cardinalities with probabilities.

Why it matters for interviews

Essential for computing probabilities of unions when events overlap. Classic interview applications include derangements, birthday problem, and coupon collector variants.

Definition and Mathematical Foundation

For the union of sets: \( |A_1 \cup \cdots \cup A_n| = \sum|A_i| - \sum|A_i \cap A_j| + \cdots + (-1)^{n+1}|A_1 \cap \cdots \cap A_n| \). Generalizes to probabilities by replacing cardinalities with probabilities.

Application in Quantitative Finance

Essential for computing probabilities of unions when events overlap. Classic interview applications include derangements, birthday problem, and coupon collector variants.

Related Terms

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Frequently Asked Questions

How is inclusion-exclusion used to count derangements?▼
A derangement is a permutation with no fixed points. Using inclusion-exclusion on the events 'element i is fixed': \( D_n = n! \sum_{k=0}^n \frac{(-1)^k}{k!} \approx n!/e \).
Is there a more efficient alternative to inclusion-exclusion?▼
For some problems, generating functions or Mobius inversion on posets generalize and simplify inclusion-exclusion. For symmetric problems, Burnside's lemma or Polya counting may be more efficient.
How does this apply to the birthday problem?▼
Computing the probability that at least two people share a birthday uses the complement: \( 1 - \frac{365!}{365^n(365-n)!} \). For exact 'exactly k collisions,' inclusion-exclusion is needed.