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 PF(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. PF(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.
- Random walk
- Stopping times