How to Gamble if You Must

Kyle Siegrist

Author Information

  • Kyle Siegrist
  • Department of Mathematical Sciences
  • University of Alabama in Huntsville


In red and black, a player bets, at even stakes, on a sequence of independent games with success probability p, until she either reaches a fixed goal or is ruined. In this article we explore two strategies: timid play in which the gambler makes the minimum bet on each game, and bold play in which she bets, on each game, her entire fortune or the amount needed to reach the target (whichever is smaller). We study the success probability (the probability of reaching the target) and the expected number of games played, as functions of the initial fortune. The mathematical analysis of bold play leads to some exotic and beautiful results and unexpected connections with dynamical systems. Our exposition (and the title of the article) are based on the classic book Inequalities for Stochastic Processes; How to Gamble if You Must, by Lester E. Dubbins and Leonard J. Savage.

Technologies Used in This Article

The XML version of this article uses the Mathematics Markup Language (MathML) for the display of mathematical expressions. You will need the Mozilla Firefox browser, either Version 2.x with the MIT MathML fonts installed, or Version 3.x with the STIX fonts installed. Interactive applets are written Java. You will need the Java plug-in (version 1.5 or later) to view these applets. Ancillary pages are displayed using JavaScript so you will need JavaScript enabled in your browser.

The PDF version of the article contains just the expository text and graphics, and should be accessible with any browser that has a PDF plug-in.

Publication Data

  • Published July 2008
  • Copyright © 2008 by Kyle Siegrist

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