# Ballot Theorems, Old and New

• B. A. Reed
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 17)

## Abstract

We begin by sketching the development of the classical ballot theorem as it first appeared in the Comptes Rendus de 1’Academie des Sciences. The statement that is fairly called the first Ballot Theorem was due to Bertrand: Theorem 1 ([8]). We suppose that two candidates have been submitted to a vote in which the number of voters is μ. Candidate A obtains n votes and is elected; candidate B obtains m = μ − n votes. We ask for the probability that during the counting of the votes, the number of votes for A is at all times greater than the number of votes for B. This probability is (2nμ)/μ = (nm)/(n + m).

## Keywords

Random Walk Random Measure Strong Markov Property Real Random Variable Fair Game
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Erik Sparre Andersen, Fluctuations of sums of random variables, Mathematica Scandinavica, 1 (1953), 263–285.
2. [2]
Erik Sparre Andersen, Fluctuations of sums of random variables ii, Mathematica Scandinavica, 2 (1954), 195–223.
3. [3]
Désiré André, Solution directe du probleme resolu par M. Bertrand, Comptes Rendus de l’Academie des Sciences, 105 (1887), 436–437.Google Scholar
4. [4]
N. Balakrishnan, Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkhäuser, Boston, MA, first edition (1997).
5. [5]
Émile Barbier, Generalisation du probleme resolu par M. J. Bertrand, Comptes Rendus de l’Academie des Sciences, 105 (1887), 407.Google Scholar
6. [6]
J. Bertrand, Observations, Comptes Rendus de l’Academie des Sciences, 105 (1887), 437–439.Google Scholar
7. [7]
J. Bertrand, Sur un paradox analogue au problème de Saint-Pétersburg, Comptes Rendus de l’Academie des Sciences, 105 (1887), 831–834.Google Scholar
8. [8]
J. Bertrand, Solution d’un probleme, Comptes Rendus de l’Academie des Sciences, 105 (1887), 369.Google Scholar
9. [9]
Aryeh Dvoretzky and Theodore Motzkin, A problem of arrangements, Duke Mathematical Journal, 14 (1947), 305–313.
10. [10]
Meyer Dwass, A fluctuation theorem for cyclic random variables, The Annals of Mathematical Statistics, 33(4) (December 1962), 1450–1454.
11. [11]
William Feller, An Introduction to Probability Theory and Its Applications, Volume 1, volume 1. John Wiley & Sons, Inc, third edition (1968).Google Scholar
12. [12]
I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, Saunders Mathematics Books. W. B. Saunders Company (1969).Google Scholar
13. [13]
Philip S. Griffin and Terry R. McConnell, On the position of a random walk at the time of first exit from a sphere, The Annals of Probability, 20(2) (April 1992), 825–854.
14. [14]
Philip S. Griffin and Terry R. McConnell, Gambler’s ruin and the first exit position of random walk from large spheres, The Annals of Probability, 22(3) (July 1994), 1429–1472.
15. [15]
Howard D. Grossman. Fun with lattice-points, Duke Mathematical Journal, 14 (1950), 305–313.Google Scholar
16. [16]
A. Hald, A History of Probability and Statistics and Their Applications before 1750, John Wiley & Sons, Inc, New York, NY (1990).
17. [17]
Olav Kallenberg, Foundations of Modern Probability, Probability and Its Applications, Springer Verlag, second edition (2003).Google Scholar
18. [18]
Olav Kallenberg, Ballot theorems and sojourn laws for stationary processes, The Annals of Probability, 27(4) (1999), 2011–2019.
19. [19]
Harry Kesten, Sums of independent random variables — without moment conditions, Annals of Mathematical Statistics, 43(3) (June 1972), 701–732.
20. [20]
Harry Kesten, Frank spitzer’s work on random walks and Brownian motion, Annals of Probability, 21(2) (April 1993), 593–607.
21. [21]
Harry Kesten and R. A. Maller, Infinite limits and infinite limit points of random walks and trimmed sums, The Annals of Probability, 22(3) (1994), 1473–1513.
22. [22]
Takis Konstantopoulos, Ballot theorems revisited, Statistics & Probability Letters, 24(4) (September 1995), 331–338.
23. [23]
Sri Gopal Mohanty, An urn problem related to the ballot problem, The American Mathematical Monthly, 73(5) (1966), 526–528.
24. [24]
Émile Rouché. Sur la durée du jeu, Comptes Rendus de l’Academie des Sciences, 106 (1888), 253–256.Google Scholar
25. [25]
Émile Rouché, Sur un problème relatif à la durée du jeu, Comptes Rendus de l’Academie des Sciences, 106 (1888), 47–49.Google Scholar
26. [26]
Frank Spitzer, A combinatorial lemma and its applications to probability theory, Transactions of the American Mathematical Society, 82(2) (July 1956), 323–339.
27. [27]
Charles J. Stone, On local and ratio limit theorems, in: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (1965), pp. 217–224.Google Scholar
28. [28]
Lajos Takács, Ballot problems, Zeitschrift für Warscheinlichkeitstheorie und verwandte Gebeite, 1 (1962), 154–158.
29. [29]
Lajos Takács, A generalization of the ballot problem and its application in the theory of queues, Journal of the American Statistical Association, 57(298) (1962), 327–337.
30. [30]
Lajos Takács, The time dependence of a single-server queue with poisson input and general service times, The Annals of Mathematical, 33(4) (December 1962), 1340–1348.
31. [31]
Lajos Takács, The distribution of majority times in a ballot, Zeitschrift für Warscheinlichkeitstheorie und verwandte Gebeite, 2(2) (January 1963), 118–121.
32. [32]
Lajos Takács, Combinatorial methods in the theory of dams, Journal of Applied Probability, 1(1) (1964), 69–76.
33. [33]
Lajos Takács, Fluctuations in the ratio of scores in counting a ballot, Journal of Applied Probability, 1(2) (1964), 393–396.
34. [34]
Lajos Takács, A combinatorial theorem for stochastic processes, Bulletin of the American Mathematical Society, 71 (1965), 649–650.
35. [35]
Lajos Takács, On the distribution of the supremum for stochastic processes with interchangeable increments, Transactions of the American Mathematical Society, 119(3) (September 1965), 367–379.
36. [36]
Lajos Takács, Combinatorial Methods in the Theory of Stochastic Processes, John Wiley & Sons, Inc, New York, NY, first edition (1967).
37. [37]
Lajos Takács, On the distribution of the maximum of sums of mutually independent and identically distributed random variables, Advances in Applied Probability, 2(2) (1970), 344–354.
38. [38]
Lajos Takács, On the distribution of the supremum for stochastic processes, Annales de l’Institut Henri Poincaré B, 6(3) (1970), 237–247.
39. [39]
J. C. Tanner, A derivation of the borel distribution, Biometrika, 48(1–2) (June 1961), 222–224.

© János Bolyai Mathematical Society and Springer-Verlag 2008