Bertrand's ballot theorem

When candidates receive p and q votes, with p greater than q, the probability that the p-vote candidate leads after every counted vote in a uniformly random ordering is (p−q)/(p+q).

Ballot sequence: A sequence of votes for two candidates, represented as steps that increase one candidate’s count at a time. Each vote order is a path whose lead must stay positive for the theorem’s event.
Combinatorics: The branch of mathematics concerned with counting, arrangement, and discrete structures. The theorem reduces a probability to counting favorable and total vote orders.
Joseph Louis François Bertrand: A French mathematician whose work ranged across probability, geometry, and mathematical physics. Bertrand published the ballot problem that gave the theorem its name.
Election forecasting: The use of data and statistical models to estimate the outcomes of elections. The theorem quantifies how often a final winner also leads at every stage under random vote order.
Bertrand's ballot problem: The counting question of how often one candidate stays ahead throughout a sequential count of votes. The theorem is the closed-form probability answering this underlying ballot problem.
Reflection principle: A counting technique that reflects paths after their first contact with a boundary to pair constrained and unconstrained paths. It provides a standard way to count vote orders that fail the persistent-lead condition.
Permutation: An ordering of the elements of a finite set. A random ballot is a permutation of the votes, with votes of the same candidate treated as indistinguishable.
William Allen Whitworth: A British mathematician known for work in probability, combinatorics, and the theory of permutations. Whitworth supplied an early proof of the ballot result after Bertrand posed the problem.
Random walk: A process that moves among states through successive random steps. The evolving margin between two candidates is a finite random walk with a fixed endpoint.
Bertrand's ballot theorem (weak form): A ballot-counting result for sequences in which the leading candidate may tie after intermediate votes. Allowing ties changes the strict-lead event used in the stated theorem.

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Bertrand's postulateCompared with: Its shared name is unrelated; it concerns ballot sequences rather than prime distribution.