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The Gambler’s Ruin problem is one of the fundamental problems in probability theory and actuarial economics. It provides a mathematical framework to study the eventual fate of a gambler playing a series of bets where he either wins or loses a fixed amount of money with certain probabilities. The basic setting involves a gambler who starts with an initial stake and repeatedly plays a game until he either becomes completely ruined, meaning his stake falls to zero, or he reaches a target fortune, which is usually a pre-specified upper bound. This problem can be modeled as a random walk with absorbing barriers at both ends, where each step corresponds to a gain or loss in wealth. In actuarial economics, the gambler’s ruin problem not only illustrates risk and ruin probabilities in games of chance but also has applications in insurance mathematics, portfolio management, and survival analysis in finance. In the simplest version of the problem, consider a gambler who starts with an initial wealth of *i* units. He plays against an opponent who has a fortune of *N – i* units, so that the total combined wealth is *N*. At each round, the gambler wins one unit with ________ __________ _______ ___ ________ _________ _____ ______ __________ _________.
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