vote for the second candidate is received move up 1 unit. Each such path corresponds to a unique sequence of votes and will end at (p, q). A sequence is 'good' exactly when the corresponding path never goes above the diagonal line y = x; equivalently, a sequence is 'bad' exactly when the corresponding path touches the line y = x + 1. For each 'bad' path P, define a new path P′ by reflecting the part of P up to the first point it touches the line across it. P′ is a path from (−1, 1) to (p, q). The same operation applied again restores the original P. This produces a one-to-one correspondence between the 'bad' paths and the paths from (−1, 1) to (p, q). The number of these paths is and so that is the number of 'bad' sequences. This leaves the number of 'good' sequences as Since there are altogether, the probability of a sequence being good is . In fact, the solutions to the original problem and the variant problem are easily related. For candidate A to be strictly ahead throughout the vote count, they must receive the first vote and for the remaining votes (ignoring the first) they must be either strictly ahead or tied throughout the count. Hence the solution to the original problem is as required. Conversely, the tie case can be derived from the non-tie case. Note that the number of non-tie sequences with p+1 votes for A is equal to the number of tie sequences with p votes for A. The number of non-tie votes with p + 1 votes for A votes is , which by algebraic manipulation is , so the fraction of sequences with p votes for A votes is . Notes References Ballot theorems, old and new, L. Addario-Berry, B.A. Reed, 2007, in Horizons of combinatorics, Editors Ervin Győri, G. Katona, Gyula O. H. Katona, László Lovász, Springer, 2008, External links The Ballot Problem (includes scans of the original French articles and English translations) Bernard Bru, Les leçons de calcul des probabilités de Joseph Bertrand, history of the problem (in French) Probability problems Enumerative combinatorics Theorems in combinatorics Probability theorems Articles containing proofs Voting theory