Combinatorics Enumerative combinatorics: sequences and numbers Theorem

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.

History

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

Reading path

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Binomial coefficient definition
Bertrand's ballot theorem statement

Tag	`0FS972T0ZH`
URI	`m2://0FS972T0ZH`
Kind	statement, theorem
Writing	W1 stated
MSC	05A15
Status	This copy of the site has no build data. Run `m2 status 0FS972T0ZH`.
Uses	Binomial coefficient
Same as	Wikidata Q2253746 Wikipedia Bertrand's ballot theorem

Bertrand's ballot theorem

Statement

Remarks

History

Reading path

Further reading

What this page needs for W2

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