Martingale and other betting systems: why they can't beat the house edge
Martingale, Paroli, D'Alembert, Fibonacci and Labouchère put to exact math and simulation: losing streaks, the bets they force and an edge that never moves.
Fact-checked 18 October 2026 · 6 sourcesA betting system tells you how much to stake next based on what just happened. It can change how a session feels, with many small wins and the occasional disaster, but it can't change the odds of a single bet, so it can't change the house edge. This article shows why, with streak probabilities, bet sizes and a head-to-head simulation.
The one fact that sinks every system
Every bet has its own expected value: minus the edge times the stake. A system only decides the stakes. The expected result of a whole session is the sum of each possible bet's expected value times the chance you end up making it, so it always comes out as minus the edge times the total you can expect to wager.
Mathematicians prove the general version as the optional stopping theorem. As long as there is any limit on the size of bets, your bankroll or your time, no rule for when to raise, lower or stop can turn bets that lose on average into a session that wins on average. All of this rests on each bet being independent of the last: past spins don't change the next one, as the gambler's fallacy article explains.
Martingale: double after every loss
The Martingale dates back to 18th-century France. Start with 1 coin on an even-money bet; after a loss, double; after a win, go back to 1. The first win recovers everything lost in the run plus 1 coin. Here is what a run of losses costs:
| Losses in a row | Last bet | Total lost | Next bet | |--:|--:|--:|--:| | 1 | 1 | 1 | 2 | | 3 | 4 | 7 | 8 | | 5 | 16 | 31 | 32 | | 8 | 128 | 255 | 256 | | 10 | 512 | 1,023 | 1,024 | | 15 | 16,384 | 32,767 | 32,768 |
All of that is risked to win 1 coin.
How often the long streak comes
On single-zero red, a bet loses 19 times in 37. The chance that a particular run starts with 6 losses is 1.83%, and with 10 losses 0.13%. That sounds rare, but a Martingale player isn't making one run; they're making hundreds. The chance of at least one run of 10 losses somewhere in a stretch of spins:
| Game, losing chance | In 200 bets | In 1,000 bets | In 5,000 bets | |---|--:|--:|--:| | Single-zero red, 19/37 | 11.3% | 46.2% | 95.6% | | 15-slot wheel T or CT, 8/15 | 15.5% | 58.1% | 98.7% | | Dice at 49.5%, 0.505 | 9.8% | 41.3% | 93.1% |
Even a run of 15 losses, the one that has already cost 32,767 coins, turns up in 5,000 single-zero spins 10.5% of the time. These come from an exact calculation that tracks the current losing run bet by bet; it reproduces the figures in Wikipedia's Martingale article (about 84% for a run of six in 200 spins, 11% for a run of ten).
The average of one cycle
Call a cycle the bets up to the first win, or up to the point where you can't double again. With room for 10 losses (1,023 coins) on single-zero red, a cycle wins 1 coin 99.87% of the time and loses 1,023 coins 0.13% of the time:
0.998725 × 1 − 0.001275 × 1,023 = −0.3056 coins per cycle.
A cycle wagers 11.308 coins on average, and −0.3056 / 11.308 = −2.70%, which is exactly the wheel's edge. Wikipedia gives the general formula for one cycle as B(1 − (2q)ⁿ), where B is the base bet, q the losing chance and n the losses you can afford. It is negative whenever q is above one half, which is the case for every even-money bet with a house edge.
Bet limits and bankroll
Doubling also hits a ceiling: every game on cs2.casino shows its bet limits. If the maximum were 1,000 times the minimum, you could double only through 1, 2, 4 … 512, and the eleventh bet of 1,024 wouldn't be allowed. A bankroll does the same job: whoever runs out of room to double takes the whole run's loss at once. The theory only works with unlimited money, unlimited bets and unlimited time.
The other systems
- Paroli, the reverse Martingale: double after a win, go back to the base bet after a loss or after three wins in a row. A cycle can only lose 1 unit, but it completes only when you win three in a row, (18/37)³ = 11.5% of the time on single-zero red. A 2026 conference paper by Peter Pflaumer of TU Dortmund describes it as the Martingale's mirror image and finds the house edge unchanged.
- D'Alembert: add 1 unit after a loss, take 1 off after a win. It chases losses more slowly than the Martingale.
- Fibonacci: stakes follow 1, 1, 2, 3, 5, 8, 13…, one step forward after a loss and two back after a win. In the Wizard of Odds' worked example, a player aiming for a $10 win had to bet $130 at one point.
- Labouchère: write a list such as ten 1s and bet the first plus the last number. A win crosses both off; a loss adds the amount lost to the end. The session is won when the list is empty.
A head-to-head test
Each system played single-zero red under the same rules: a 100-coin bankroll and a 1-coin base, stopping at 10 coins up, at zero or after 1,000 bets, with any bet larger than what's left cut to what's left. The Martingale was computed exactly; the others were simulated over 4 million sessions each, with a fixed-seed random number generator so the run can be repeated.
| System | Ended ahead | Lost all 100 | Lost per 100 coins wagered | |---|--:|--:|--:| | Martingale (exact) | 88.9% | 11.1% | 2.703 | | Paroli | 73.7% | 15.8% | 2.704 | | D'Alembert | 86.3% | 13.7% | 2.700 | | Fibonacci | 87.9% | 12.1% | 2.694 | | Labouchère | 88.0% | 12.0% | 2.685 | | Flat 1-coin bets | 53.7% | 1.5% | 2.701 |
The systems that chase losses end ahead most often and pay for it with the one session in seven to nine that loses everything. Paroli never bets more than 4 coins, yet it still lost the whole bankroll in 15.8% of sessions, through a steady drip of small losses over hundreds of bets. Flat betting lost more per session on average only because its sessions ran longer and wagered more. Per coin wagered, every row lands on the wheel's 2.703% edge, within simulation noise. Run again on a dice bet at a 1% edge, every system lost between 0.96 and 1.03 coins per 100 wagered.
The Wizard of Odds' long simulations of D'Alembert, Fibonacci and Labouchère show the same thing across baccarat, craps and both roulette wheels: money lost divided by money bet equals the game's edge, whatever the bankroll.
Auto-bet is the same math, faster
If a game offers auto-bet with a rule like "on loss, increase the stake by 100%", that rule is a Martingale run by a script; "on win, increase by 100%" is a Paroli. Automation doesn't change any expected value. It changes the speed: more bets an hour means more wagered an hour, and the long losing streak arrives sooner on the clock.
If you use it, decide the stopping point before you start, not in the middle of a run. On cs2.casino, deposit, loss and wager limits under your account limits can cap a rolling day, week or month; lowering a limit applies at once, while raising one waits out a cooling-off period. The responsible gambling guide covers the tools in more detail.
What actually changes the cost
The expected loss is the edge times the total wagered. The only things that move it are the edge of the game you choose, the size of your stakes, how many bets you make and when you stop. A system that usually wins a little is a bet on dodging a rare large loss, and the averages above show what that bet is worth. The house edge article works out the edge for the common games.
