### Abstract

Let prize X in a game be a random variable with a cumulative distribution function F, E[X] ≠ 0, and Var(X) < ∞. In a Gambler's Ruin Problem we consider the probability P_{F}(A, B) of accumulating fortune A before losing the initial fortune B. Suppose our Gambler is to choose between different strategies with the same expected values and different variances. P_{F}(A, B) is known to depend in general on the whole cumulative distribution function F of X. In this paper we derive an approximation which implies the following rule called A Rule of Thumb (not only) for Gamblers: if E(X) < 0 then the strategy with the greater variance is superior, while in case E[X] > 0 the strategy with the smaller variance is more favorable to the Gambler. We include some examples of applications of The Rule. Moreover we derive a general solution in the Roulette case and use it to show good behavior of The Rule explicitly.

Original language | English |
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Pages (from-to) | 169-181 |

Number of pages | 13 |

Journal | Stochastic Processes and their Applications |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1995 |

### Keywords

- Gambling
- Martingale
- Random walk
- Ruin
- Stopping times

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## Cite this

*Stochastic Processes and their Applications*,

*55*(1), 169-181. https://doi.org/10.1016/0304-4149(95)91546-D