Hostname: page-component-cb9f654ff-plnhv Total loading time: 0.001 Render date: 2025-08-21T15:06:50.425Z Has data issue: false hasContentIssue false
  • English
  • Français

More on n-point, win-by-k games

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

John Haigh
John Haigh*
Affiliation:
University of Sussex
*
∗ Postal address: School of Mathematical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

When Siegrist (1989) derived an expression for the probability that player A wins a game that consists of a sequence of Bernoulli trials, the winner being the first player to win n trials and have a lead of at least k, he noted the desirability of giving a direct probabilistic argument. Here we present such an argument, and extend the domain of applicability of the results beyond Bernoulli trials, including cases (such as the tie-break in lawn tennis) where the probability of winning each trial cannot reasonably be taken as constant, and to where there is Markov dependence between successive trials.

Keywords

BERNOULLI TRIALSGAMBLER'S RUINPAIRED COMPARISONSMARKOV CHAINS

MSC classification

Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 382 - 387
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Feller, W. (1969) An Introduction to Probability Theory and its Applications. Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Mohan, C. (1955) The gambler's ruin problem with correlation. Biometrika 42, 486–193.CrossRefGoogle Scholar
Proudfoot, A. D. and Lampard, D. G. (1972) A random walk problem with correlation. J Appl. Prob. 9, 436–440.CrossRefGoogle Scholar
Renshaw, E. and Henderson, R. (1981) The correlated random walk. J. Appl. Prob. 18, 403–414.CrossRefGoogle Scholar
Siegrist, K. (1989) n-point, win-by-k-games. J. Appl. Prob. 26, 807–814.CrossRefGoogle Scholar