In combinatorics, Bertrand's ballot theorem is the solution to the question: "In an election where one candidate receives p votes and the other q votes with p≥q, what is the probability that the first candidate will be strictly ahead of the second candidate throughout the count?" The answer is Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ... An election is a process in which a vote is held to choose amongst candidates to fill an office, or amongst political parties offering a slate of potential office holders for a house of representatives. ... The word probability derives from the Latin probare (to prove, or to test). ...

(p−q)/(p+q).

It is related to random walks and can be proved several different ways. One is by mathematical induction: In mathematics and physics, a random walk is a formalization of the intuitive idea of taking successive steps, each in a random direction. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ...

• Clearly it is true if p>0 and q=0 when the probability is 1, given that the first candidate receives all the votes; it is also true when p=q>0 since the probability is 0, given that the first candidate will not be strictly ahead after all the votes have been counted.
• Assume it is true both when p=a−1 and q=b, and when p=a and q=b−1, with a>b>0. Then considering the case with p=a and q=b, the last vote counted is either for the first candidate with probability a/(a+b), or for the second with probability b/(a+b). So the probability of the first being ahead throughout the count to the penultimate vote counted (and also after the final vote) is:

• And so it is true for all p and q with p>q>0.

It can then be used to calculated the number of one-dimensional walks of n steps from the origin to the point m which do not return to the origin. Assuming n and m have the same parity and n≥m>0, it is When m=1 and n is odd, this gives the Catalan numbers. The Catalan numbers, named after the Belgian mathematician Eugène Charles Catalan (1814—1894), form a sequence of natural numbers that occur in various counting problems in combinatorics. ...