Suppose that in a ballot candidate A scores a votes and candidate B scores 6 votes and that all the possible voting records are equally probable. Let a, and p, denote the ilumber of votes registered for A and B among the first r votes recorded. Let A (A*) denote the, number of subscripts r for which the inequalitt (strict inequaiity) a r 2 p& + u (a, > p& + u) holds where ~12 0 and v are arbitrary real numbers. Simple combinatorial proofs are given for the following theorems of which the first two are due to Takdcs [4,5] and the third is new. (1) If ca and b are relatively prime, ~1 = a/b and v = 0, then P(d =j)= l/(a + b), i = 1, 2,. . . , 6. p= (a -1)/b P(A=j)=l/(a+b),j=O,..., a + 6 -1. Nso, closed form expressions for #the probabillities P(A =a+& and P(A*= a + 6) are given without any restrictions on CC and v special cases of which have been derived by Takfics [6] among others, Suppose that in a ballot candidate A scores d votes and candidate I$ scores 6 votes and that all the possible (*,'") voting records are equally probable, Denote by cy, and & the number of votes registered for A and I3 among the first r votes