aez-notes

In an election, two candidates, Albert and Benjamin, have in a ballot box \(a\) and \(b\) votes respectively, \(a > b\), for example, 3 and 2. If ballots are randomly drawn and tallied, what is the chance that at least once after the first tally the candidates have the same number of tallies?

Observe first that if the first vote drawn is a \(B\), which occurs with probability \(b / (a + b)\), then they will definitely have the same number of votes at some point in the future.

There are the same number of trajectories starting from that first vote that reach parity as there are trajectories that reach partity if the first ballot drawn is an \(A\). Since all trajectories are equiprobable, the probability of getting back to parity after a first \(A\) is the same as the probability of getting a \(B\) first and then getting back to parity (since the trajectories are in bijection).

Therefore, the probabilility is \(2b / (a + b)\).