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

Session Title

Poster session

Presentation Type

Poster Presentation

Start Date

27-5-2026 12:00 AM

Abstract

This study provides a comprehensive analysis of Paroli (reverse Martingale) and Let-It-Ride betting systems in Roulette, combining analytical formulas, recursive sequence analysis, and Monte Carlo simulations to quantify expected gain, variance, and normalized risk measures. We develop exact expressions for net gain, total wager, and associated risk metrics, enabling systematic performance comparisons across sequences and bet sizes. A key theoretical contribution is the demonstration that Paroli forms a mathematical mirror of the classical Martingale system: their performance formulas are related by swapping win and loss probabilities. In the fair-game case, Paroli and Martingale are statistically equivalent, with identical moments, differing only in whether bets increase after wins or losses. We extend this framework to Let-It-Ride variants, analyzing covariances and correlation coefficients between gain and total wager to characterize each system's risk structure. 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 house edge remains unchanged across all systems. Normalized risk measures reveal substantial differences in volatility, with Paroli exhibiting higher gain-wager correlation than the classical Martingale under typical casino conditions.

Author Bios

Peter Pflaumer is a retired professor of statistics and demography. His research covers population statistics and forecasting, historical demography, time series analysis, financial mathematics, and investment analysis. He has published extensively on population forecasting, ARIMAX models, investment calculations, and gambling systems like the roulette Martingale.

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May 27th, 12:00 AM

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

This study provides a comprehensive analysis of Paroli (reverse Martingale) and Let-It-Ride betting systems in Roulette, combining analytical formulas, recursive sequence analysis, and Monte Carlo simulations to quantify expected gain, variance, and normalized risk measures. We develop exact expressions for net gain, total wager, and associated risk metrics, enabling systematic performance comparisons across sequences and bet sizes. A key theoretical contribution is the demonstration that Paroli forms a mathematical mirror of the classical Martingale system: their performance formulas are related by swapping win and loss probabilities. In the fair-game case, Paroli and Martingale are statistically equivalent, with identical moments, differing only in whether bets increase after wins or losses. We extend this framework to Let-It-Ride variants, analyzing covariances and correlation coefficients between gain and total wager to characterize each system's risk structure. 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 house edge remains unchanged across all systems. Normalized risk measures reveal substantial differences in volatility, with Paroli exhibiting higher gain-wager correlation than the classical Martingale under typical casino conditions.