Bertrand's Ballot Theorem - Proof By Reflection

Proof By Reflection

For A to be strictly ahead of B throughout the counting of the votes, there can be no ties. Separate the counting sequences according to the first vote. Any sequence that begins with a vote for B must reach a tie at some point, because A eventually wins. For any sequence that begins with A and reaches a tie, reflect the votes up to the point of the first tie (so any A becomes a B, and vice-versa) to obtain a sequence that begins with B. Hence every sequence that begins with A and reaches a tie is in one to one correspondence with a sequence that begins with B, and the probability that a sequence begins with B is, so the probability that A always leads the vote is

the probability of sequences that tie at some point

the probability of sequences that tie at some point and begin with A or B