The promise of a progression
A staking progression changes the size of your next bet according to what just happened. The best known is the Martingale: bet 1 unit, and after each loss double the stake; after a win return to 1 unit. The attraction is easy to see. Because a win after any run of losses recovers all earlier losses and leaves a profit of one unit, it feels as if you only need one win.
The title promises a way to overcome the house edge. It cannot be done, and the arithmetic below shows why. This article is about understanding that, not about a method.
What a doubling sequence looks like
Start at 1 unit on an even-money bet and double after every loss:
| Bet number | Stake | Total staked so far |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 3 |
| 3 | 4 | 7 |
| 4 | 8 | 15 |
| 5 | 16 | 31 |
| 6 | 32 | 63 |
| 7 | 64 | 127 |
The pattern is that after k losses in a row you have lost 2^k - 1 units in total, and the next stake is 2^k. If the next bet wins, you receive 2^k and your net result across the whole sequence is +1. For example, losing 1, 2 and 4 (7 units) and then winning with a stake of 8 returns 16, leaving a net gain of 16 - 15 = 1.
So a win always nets one unit. That is the only profit the system can make from a sequence, however deep it goes.
How likely is a long losing run?
Take a single zero roulette wheel and an even-money bet such as red. The chance of losing a given spin is 19/37 (18 black plus the zero), about 0.5135.
- Chance of losing 6 in a row: (19/37)^6, about 0.0183, or roughly 1.8%. That is about 1 in 55.
- Chance of losing 9 in a row: (19/37)^9, about 0.0025, roughly 1 in 400.
These are not remote. If someone started many such sequences, a run of six losses would be something they could expect to meet from time to time. And in the sequence above, six losses in a row means 63 units lost with the seventh stake of 64 still to come.
Putting numbers on a sequence
Suppose, as an illustration, a player follows the doubling plan for at most 6 bets and then stops, accepting the loss if all 6 fail.
- Chance of winning at some point in the 6 bets: 1 - 0.0183 = 0.9817, about 98.2%. The net result is +1 unit.
- Chance of losing all six: 0.0183, about 1.8%. The net result is -63 units.
Expected result per round: 0.9817 × 1 - 0.0183 × 63 = 0.9817 - 1.1529, which is about -0.17 units.
(Using the unrounded figures, 0.0183 is about 0.018337, and the result is about -0.1736, so -0.17 is a fair rounding.)
The pattern is typical of progressions: very frequent small gains, rare large losses, and an average that is negative. The negative average is the same house edge as before. A cross-check: the expected total stake per round works out to roughly 6.42 units, and 6.42 × 1/37 is about 0.17, so the loss is the 1/37 edge applied to everything staked. The system did not remove the edge; it just made the player stake more in total, so the edge applied to a larger amount.
The real limits: tables and bankrolls
Real play adds two constraints on top of the maths.
Table limits. Tables have maximum stakes. As an illustration, if the maximum were 500 and the starting stake 1, the stakes would run 1, 2, 4, 8, 16, 32, 64, 128, 256. The next stake, 512, exceeds 500 and would not be allowed. After those nine losses you would have lost 511 units with no way to continue the doubling. The chance of nine consecutive losses at 19/37 is about 0.25%. Limits vary between tables, but every table has one.
Finite funds. The stakes double each time, so the money needed grows fast, and basic bankroll maths shows how quickly. To survive six losses and place the seventh bet you need 127 units at the start, to win 1. That ratio of risked to won is why a tiny gain is not worth the exposure. This is the gamblers ruin problem: with finite funds, a long enough unlucky run ends the game.
Other progressions
Variants, such as increasing the stake by a fixed amount after a loss or following a list of numbers, change the shape of the wins and losses. They do not change the key fact that each bet has a negative expected value, so the total expected result is the sum of negative numbers. Reducing the size of the bets reduces how quickly you lose on average; no plan turns it positive.
Why they feel as if they work
- Frequent wins. Most sequences end in a small profit, so a player usually sees success.
- Long gaps between disasters. The big loss may not happen for a long time, which feels like proof.
- The gambler’s fallacy. After several losses, a win seems due. Each spin is independent, and the wheel has no memory.
Because wins are common and losses are large and rare, the experience misleads. Only the average over a great many sequences shows the truth.
Important reminders
Gambling can cause serious harm. No system makes it safe, results are uncertain, and a plan that raises stakes after losses can lead to big losses quickly. Never borrow to play, never bet more than you can afford to lose and avoid chasing losses. If gambling is a problem for you or someone you know, contact a local support service.