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Bertrand's ballot theorem - mklabgr - 06-22-2026
Bertrand's ballot theorem
BY WIKIPEDIA
Summary
Bertrand’s ballot theorem is a classic result in combinatorics and probability that studies the following question: if candidate A receives (p) votes and candidate B receives (q) votes with ($p>q$), what is the probability that A is always strictly ahead during the entire vote count when the ballots are counted in a random order? The theorem states that this probability is ($\frac{p-q}{p+q}$). For example, if A wins 3–2, the chance that A remains ahead at every stage of the count is ($\frac{3-2}{3+2}=\frac15$).
The theorem, first discovered by W. A. Whitworth and later rediscovered by Joseph Bertrand, has elegant proofs using methods such as reflection, induction, and the cycle lemma, and it is an important example connecting probability, random walks, lattice paths, and combinatorial counting.
ARTICLE
Four Proofs of the Ballot Theorem BY MARC RENAULT