Ballot Theorems, Old and New

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We begin by sketching the development of the classical ballot theorem as it first appeared in the Comptes Rendus de 1’Academie des Sciences. The statement that is fairly called the first Ballot Theorem was due to Bertrand: Theorem 1 ([8]). We suppose that two candidates have been submitted to a vote in which the number of voters is μ. Candidate A obtains n votes and is elected; candidate B obtains m = μ − n votes. We ask for the probability that during the counting of the votes, the number of votes for A is at all times greater than the number of votes for B. This probability is (2nμ)/μ = (nm)/(n + m).