Analyzing Paroli and Let-It-Ride Betting Systems: Expected Gain, Variance, and Risk Measures in Roulette

Abstract

This study presents a comprehensive analysis of the Paroli (reverse Martingale) and Let-It-Ride betting systems in roulette, combining exact analytical formulas, recursive sequence methods, and Monte Carlo simulations to quantify expected gain, variance, and normalized risk measures. We derive closed-form expressions for net gain, total wager, and associated risk metrics, enabling systematic comparisons across sequences and bet sizes. A key theoretical insight is that Paroli constitutes a mathematical mirror of the classical Martingale system: their performance formulas are related by swapping win and loss probabilities. In the fair-game limit, Paroli and Martingale are statistically equivalent, with identical moments, differing only in whether bets increase after wins or losses. Extending this framework to Let-It-Ride variants, we analyze covariances and correlations between gain and total wager to characterize the risk structure of each system. Sequence-based tables illustrate outcome distributions, while simulations validate analytical results and highlight comparative volatility. Our findings confirm that, although Paroli and Let-It-Ride strategies can generate large individual gains, the underlying house edge remains unchanged.