Bertrand's Ballot Theorem

Bertrand's ballot theorem, sometimes called the ballot problem, is a result in combinatorics that answers a question about vote counting. It asks, in an election where candidate A receives p votes and candidate B receives q votes with A ahead of B overall, what is the probability that A remains strictly ahead of B throughout the entire count if the votes are tallied in a random order. The answer is the fraction formed by p minus q divided by p plus q. The result was first published by W. A. Whitworth in 1878, but is named for Joseph Louis François Bertrand, who rediscovered it in 1887; a more direct proof was later given by Desire Andre using a technique now popularly known as the reflection method, even though Andre own argument did not actually involve any reflections.