The Martingale is the oldest betting system in the casino, popular in 18th-century France and still the first one most players think of. You bet one unit on an even-money bet such as red. If you lose, you double the bet. When you finally win, you are one unit up, and you start again. It looks unbeatable because you win almost every run. The catch is the rare run you cannot finish: a long enough losing streak forces a bet that is above the table limit or beyond your money, and that one loss wipes out hundreds of small wins.
How a run works
| Bet number | Bet | Total staked so far | Result if this bet wins |
|---|---|---|---|
| 1 | 1 | 1 | +1 |
| 2 | 2 | 3 | +1 |
| 3 | 4 | 7 | +1 |
| 4 | 8 | 15 | +1 |
| 5 | 16 | 31 | +1 |
| 6 | 32 | 63 | +1 |
| 7 | 64 | 127 | +1 |
| 8 | 128 | 255 | +1 |
| 9 | 256 | 511 | +1 |
Every win, at any step, ends the run one unit up. With a table maximum of 500, the tenth bet would be 512, which is not allowed. After nine losses in a row the run is over and you are 511 units down.
How likely is the long losing streak?
On a European wheel, an even-money bet loses on 19 of the 37 pockets: the 18 numbers of the other colour plus the zero. That is a 51.35% chance of losing each spin. The chance of losing every bet in a run depends on how many doublings the table maximum allows:
| Table maximum (first bet 1) | Bets in a run | Chance a run fails | Cost of a failed run | Average result per run |
|---|---|---|---|---|
| 50 | 6 | 1.83% (about 1 in 55) | 63 | −0.17 |
| 500 | 9 | 0.25% (about 1 in 403) | 511 | −0.27 |
| 1,000 | 10 | 0.13% (about 1 in 784) | 1,023 | −0.31 |
A run fails rarely, but when it does it costs hundreds of runs' worth of winnings. The last column is the average of both outcomes: one unit won with the probability of success, minus the cost of a failed run with the probability of failure. It is negative at every table limit.
Worked example: Table maximum 500. A successful run wins 1 unit, with probability 99.75%. A failed run loses 511 units, with probability 0.25%. On average: 0.9975 × 1 − 0.0025 × 511 ≈ −0.27 units per run.
Why doubling cannot beat the edge
Every single bet in the run has a negative expected value: −2.70% of the stake on a European wheel. Adding up negative bets cannot produce a positive total, however you size them. Doubling only changes the shape of your results: many small wins and a few very large losses, instead of a steady drift down. Mathematicians prove this with the optional stopping theorem: with any limit on bet size or bankroll, no stopping rule turns a losing game into a winning one.
Key point: The Martingale does not reduce the house edge. It trades frequent small wins for rare large losses, and the large losses are big enough to make the total negative.
Streaks are more common than they feel
Six losses in a row sounds unlikely, and in any six given spins it is: 1.8%. But over a longer session the question is whether such a streak happens somewhere. On a single-zero wheel there is about an 84% chance of at least one streak of six losses during 200 spins. For a streak of nine, the one that breaks a run at a table maximum of 500, the chance is about 21% in 200 spins, 45% in 500 spins and 70% in 1,000 spins.
Variations do not fix it
- The Grand Martingale doubles and adds one unit after each loss. Successful runs win more, but the bets grow faster, so the table limit is reached sooner and failed runs come more often.
- The reverse Martingale doubles after wins instead of losses. It changes the shape of the results in the other direction; the expected value of every bet is still negative.
No staking system changes the house edge, because the edge is built into the payouts of each individual bet.
See what a doubling run looks like with your own table limit in the roulette calculator.
- The Martingale wins most runs, but each failed run costs as much as hundreds of successful ones.
- A table maximum of 500 allows nine doublings; losing all nine has a 0.25% chance per run.
- Over 1,000 spins, a streak of nine losses has about a 70% chance of appearing at least once.
- No staking system changes the house edge.
Sources
- Martingale (betting system) — Wikipedia: history, streak probabilities and the optional stopping theorem
- Roulette — Wikipedia: the single-zero wheel and its house edge
18+ only. No betting system changes the house edge. See our Responsible Gambling page.
RTP Data explains the mathematics of casino games, with every figure sourced or worked out in the text.