A Statistical Analysis of the Roulette Martingale System: Examples, Formulas and Simulations with R

Abstract

Some gamblers use a martingale or doubling strategy as a way of improving their chances of winning. This paper derives important formulas for the martingale strategy, such as the distribution, the expected value, the standard deviation of the profit, the risk of a loss or the expected bet of one or multiple martingale rounds. A computer simulation study with R of the doubling strategy is presented. The results of doubling to gambling with a constant sized bet on simple chances (red or black numbers, even or odd numbers, and low (1–18) or high (19–36) numbers) and on single numbers (straight bets) are compared. In the long run, a loss is inevitable because of the negative expected value. The martingale strategy and the constant bet strategy on a single number are riskier than the constant bet strategy on a simple chance. This higher risk leads, however, to a higher chance of a positive profit in the short term. But on the other hand, higher risk means that the losses suffered by doublers and by single number bettors are much greater than that suffered by constant bettors.