In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be strictly ahead of B throughout the count?" The answer is
The result was first published by W. A. Whitworth in 1878, but is named after Joseph Louis François Bertrand who rediscovered it in 1887.
In Bertrand's original paper, he sketches a proof based on a general formula for the number of favourable sequences using a recursion relation. He remarks that it seems probable that such a simple result could be proved by a more direct method. Such a proof was given by Désiré André, based on the observation that the unfavourable sequences can be divided into two equally probable cases, one of which (the case where B receives the first vote) is easily computed; he proves the equality by an explicit bijection. A variation of his method is popularly known as André's reflection method, although André did not use any reflections.
Bertrand's ballot theorem is related to the cycle lemma. They give similar formulas, but the cycle lemma considers circular shifts of a given ballot counting order rather than all permutations.
Example
Suppose there are 5 voters, of whom 3 vote for candidate A and 2 vote for candidate B (so p = 3 and q = 2). There are ten equally likely orders in which the votes could be counted: AAABB AABAB ABAAB BAAAB AABBA ABABA BAABA ABBAA BABAA BBAAA For the order AABAB, the tally of the votes as the election progresses is:
For each column the tally for A is always larger than the tally for B, so A is always strictly ahead of B. For the order AABBA the tally of the votes as the election progresses is:
For this order, B is tied with A after the fourth vote, so A is not always strictly ahead of B. Of the 10 possible orders, A is always ahead of B only for AAABB and AABAB. So the probability that A will always be strictly ahead is
and this is indeed equal to as the theorem predicts.
Equivalent problems
Favourable orders
Rather than computing the probability that a random vote counting order has the desired property, one can instead compute the number of favourable counting orders, then divide by the total number of ways in which the votes could have been counted. (This is the method used by Bertrand.) The total number of ways is the binomial coefficient ; Bertrand's proof shows that the number of favourable orders in which to count the votes is (though he does not give this number explicitly). And indeed after division this gives .
Random walks
Another equivalent problem is to calculate the number of random walks on the integers that consist of n steps of unit length, beginning at the origin and ending at the point m, that never become negative. As n and m have the same parity and , this number is
When and is even, this gives the Catalan number . Thus the probability that the a random walk is never negative and returns to origin at time is . By Stirling's formula, when , this probability is .
[Note that have the same parity as follows: let be the number of "positive" moves, i.e., to the right, and let be the number of "negative" moves, i.e., to the left. Since and , we have and . Since and are integers, have the same parity]
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 the probability of sequences that tie at some point and begin with B the probability that a sequence begins with B