The ballot problem

Question

In an election, candidate A receives a votes and candidate B receives b votes, with a b. The ballots are counted one at a time in uniformly random order. What is the probability that A is strictly ahead throughout the entire count?

Accepted Answer

The answer is the famous ballot formula P = \frac{a - b}{a + b}. Proof by reflection. View the count as a lattice path: +1 for each A vote, -1 for each B. All \binom{a+b}{a} vote orders are equally likely. We count paths that stay strictly positive after the start. - A good path must start with an A vote. There are \binom{a+b-1}{a-1} paths starting with A and \binom{a+b-1}{a} starting with B; note every path starting with B dips to -1 and must return through 0 to end at a - b 0. - Among paths starting with an A vote, pair up the bad ones (those that later touch 0) with paths starting with a B…

The ballot problem

Your answer

More Probability questions