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How to Gamble if You Must
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.
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