---
tag: 0FS972T0ZH
kind: statement
role: theorem
name: Bertrand's ballot theorem
msc: [05A15]
uses: [092GEF38YD]
sources:
  - url: https://en.wikipedia.org/wiki/Bertrand's_ballot_theorem
    locator: lead section
same_as:
  wikidata: Q2253746
  wikipedia: Bertrand's ballot theorem
---

## Statement

Let $p$ and $q$ be natural numbers with $p > q$.
In an election, candidate $A$ gets $p$ votes and candidate $B$ gets $q$ votes.
The votes are counted one at a time, in an order chosen uniformly at random among all orders.
Then the probability that $A$ is strictly ahead of $B$ after each vote is counted is
$$\frac{p - q}{p + q}.$$

## Remarks

Equivalently, the number of such counting orders, as words with $p$ letters $A$ and $q$ letters $B$, is $\frac{p - q}{p + q} \binom{p + q}{p}$.
Example: for $p = 3$ and $q = 1$, the probability is $\frac{1}{2}$. Of the $4$ orders, $AAAB$ and $AABA$ work.
A short proof uses the reflection principle for [[lattice paths|040WY0TT2G]].

## History

William Allen Whitworth published the result in 1878.
Joseph Bertrand found it again in 1887, and the theorem has his name.

## Explanations

### L1 human

If the winner gets many more votes than the loser, the winner is likely to lead the whole time.
The formula makes this precise: the chance of an unbroken lead is the margin divided by the total number of votes.