Bertrand's ballot theorem, sometimes called the ballot problem, is a result in combinatorics that answers a question about vote counting. It asks, in an election where candidate A receives p votes and candidate B receives q votes with A ahead of B overall, what is the probability that A remains strictly ahead of B throughout the entire count if the votes are tallied in a random order. The answer is the fraction formed by p minus q divided by p plus q. The result was first published by W. A. Whitworth in 1878, but is named for Joseph Louis Francois Bertrand, who rediscovered it in 1887; a more direct proof was later given by Desire Andre using a technique now popularly known as the reflection method, even though Andre own argument did not actually involve any reflections.
Facts
StatementIf candidate A receives p votes and candidate B receives q votes in an election where A finishes ahead of B, so that p is greater than q, then the probability that A remains strictly ahead of B throughout the count, when the ballots are counted in random order, equals p minus q divided by p plus q. 1 Proof YearFirst published by W. A. Whitworth in 1878. The theorem is named for Joseph Bertrand, who rediscovered it independently in 1887. Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Source Bertrand's Ballot Theorem (Wikipedia)
Sources
1. Bertrand's Ballot Theorem (Wikipedia)
Wikimedia FoundationLead section, closing sentence
The result was first published by W. A. Whitworth in 1878, but is named after Joseph Louis François Bertrand who rediscovered it in 1887.
In Branch: Combinatorics, Lead sentence
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B re
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