Suppose that in a ballot candidate A scores a votes and candidate B scores b votes, and that all the possible voting records are equally probable. Corresponding to the first r votes, let αr and βr be the numbers of votes registered for A and B, respectively. Let ρ be an arbitrary positive real number. Denote by δ(a,b,ρ)[δ*(a,b,ρ)] the number of values of r for which the inequality αr≥ρβr[αr>ρβr], r=1,...,a+b, holds. Heretofore the probability distributions of δ and δ* have been derived for only a restricted set of values of a,b, and ρ, although, as pointed out here, they are obtainable for all values of (a,b,ρ) by using a result of Takács. In this paper the authors present a derivation of the distribution of δ[δ*] whose development, for any (a,b,ρ), leads to both necessary and sufficient conditions for δ[δ*] to have a discrete uniform distribution.
