# The Martingale Betting System: What the Math Says About the Risk of Ruin

Cite as: Gorak, R. (2026). The Martingale Betting System: What the Math Says About the Risk of Ruin. Tradicted. https://www.tradicted.com/research/pflaumer-martingale-2019/
Paper: Peter Pflaumer — *A Statistical Analysis of the Roulette Martingale System*
Published in: UNLV Gaming Research and Review Journal (2019)
Original: https://oasis.library.unlv.edu/cgi/viewcontent.cgi?article=1630&context=gaming_institute

Key finding: After 10,000 simulated martingale rounds (20,529 coups) on a simple chance in European roulette (p=19/37), the expected total profit is E(W) = -3,056.27 and the probability of a positive profit is only 18.28 percent.

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The martingale, or doubling, betting system doubles the bet after every loss so a win recovers all prior losses plus the original stake. Pflaumer (2019) titles this paper "A Statistical Analysis of the Roulette Martingale System: Examples, Formulas and Simulations with R". A simulation of 10,000 martingale rounds (20,529 coups) on European roulette (p=19/37) found an expected total profit of E(W) = -3,056.27. The probability of a positive profit was only 18.28 percent using the binomial distribution.

## What the Study Found

Without a table limit, the expected number of coups per martingale round is 2.0555 and the variance is 2.17. A 1,000-run simulation of 20,529 coups produced a mean total profit of -2,948.206 (sd 3,693.94); 80 percent of runs ended negative. After 10,000 rounds, the martingale, simple-chance, and single-number strategies all shared an expected total profit of E(W) = -3,056. The standard deviation was 788.9 for simple chance, 3,654.7 for the martingale, and 4,607.1 for single-number betting. Over 100 martingale rounds, the probability of winning 100 units was 88 percent, versus 48 percent for a straight (single-number) bettor achieving a positive profit.

## Methodology

The analysis combines closed-form probability derivations with a Monte Carlo simulation written in R for the European roulette wheel (37 numbers, single zero). The primary simulation ran 20,529 coups per repetition, repeated 1,000 times, corresponding to about 10,000 martingale rounds per run. The paper reports no calendar time period, as results derive from simulation and closed-form probability formulas. Key controls were a table limit of n=10 rounds (512 units), an initial bet of one unit, and a loss probability of p=19/37.

## Key Statistics

| Metric | Finding | Context |
|---|---|---|
| Expected total profit (10,000 rounds) | E(W) = -3,056.27 | Martingale strategy, 20,529 coups, p=19/37 |
| Probability of positive profit | 18.28% (binomial distribution) | After 10,000 martingale rounds |
| Standard deviation of total profit | σ(W) = 3,654.66 | After 10,000 martingale rounds |
| Expected gain of one martingale round | E(G) = 1 - 2p^n | p = probability of losing, n = table-limit coup cap |
| Probability of losing an entire round | P = p^n | Probability the gambler loses all n bets in a round |
| Standard deviation by strategy | 788.9 (simple chance), 3,654.7 (martingale), 4,607.1 (single number) | 10,000 rounds / 20,529 coups |

## Why This Matters

The martingale system illustrates a core risk-management principle: raising bet size after a loss increases short-term win odds without changing the game's negative expected value. Traders apply the same logic when averaging down or doubling position size after a losing trade, improving recovery odds while sharply increasing potential losses. The martingale system requires unlimited capital and no table or position limits to work safely, making it a textbook example of gambler's ruin. Higher-variance strategies raise both the chance of a short-term win and the severity of worst-case losses, a trade-off central to position sizing. The failure mode is a finite account meeting a doubling sequence. A [risk of ruin calculator](/tools/risk-of-ruin-calculator/) shows the blowup probability at a given bet fraction before you commit capital to one.

## FAQ

### What is the martingale betting system?

One unit is the initial bet in Pflaumer's (2019) martingale system, doubled after every loss until a win occurs. Doubling is capped at a table limit of n=10 rounds (512 units), and the win recovers all prior losses plus the original stake.

### How likely is a martingale bettor to profit after 10,000 rounds?

18.28 percent is the probability of a positive total profit after 10,000 martingale rounds (20,529 coups) on a European roulette wheel. That figure uses the binomial distribution; the normal approximation instead gives roughly 20 percent, per Pflaumer's (2019) European roulette simulation.

### How does the martingale system compare to constant bet sizing for risk?

4,607.1 is the standard deviation of total profit for single-number (plein) betting after 10,000 martingale rounds (20,529 coups). The martingale strategy's standard deviation is 3,654.7, and simple-chance betting's is only 788.9, making single-number betting the riskiest of the three strategies.

### What is the gambler's ruin problem in the context of martingale betting?

78,026 martingale rounds (about 160,182 expected coups) are required before the probability of a positive profit falls to just 1 percent. Pflaumer (2019) uses this steep decline to illustrate how repeated negative-expectation betting drives a gambler toward eventual ruin over time.

## Source

Peter Pflaumer. (2019). A Statistical Analysis of the Roulette Martingale System: Examples, Formulas and Simulations with R. International Conference on Gambling & Risk Taking.

[Read the full paper →](https://oasis.library.unlv.edu/cgi/viewcontent.cgi?article=1630&context=gaming_institute)