On Fluctuations in Coin-Tossing

summary:Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(\matrix {a+b} \\ a \endmatrix \right)$ voting sequences are equally probable. Denote by $\alpha_r$ and by $\beta_r$ the number of votes registered for $A$ and for $B$, respectively, among the first $r$ votes recorded, $r=1, \dots, a+b$. The purpose of this paper is to derive, for $a\geq b-c$, the probability distributions of the random variables defined as the number of subscripts $r=1, \dots, a+b$ for which (i) $\alpha_r=\beta_r-c$, (ii) $\alpha_r=\beta_r-c$ but $\alpha_{r-1}=\beta_{r-1}-c\pm 1$, (iii) $\alpha_r=\beta_r-c$ but $\alpha_{r-1}=\beta_{r-1}-c\pm 1$ and $\alpha_{r+1}=\beta_{r+1}-c\pm 1$, where $c=0,\pm 1, \pm 2, \dots$