Martingale: the doubling trap in numbers
Double the bet after every loss, go back to the base bet after a win. The Martingale promises a small profit that “cannot fail”. Here is what it does at a 2x target, where it is most popular.
How it works at 2x
Every series ends at the first win, and every won series earns exactly one base bet: the win pays back all the losses in the series plus the base bet. With a base bet of 1 the bets go 1, 2, 4, 8… and ten losses in a row mean 1,023 lost to win 1.
| Round in the series | Bet | Total lost if this one loses too | Chance of losing every round so far |
|---|---|---|---|
| 1 | 1 | 1 | 51.5% |
| 2 | 2 | 3 | 26.5% |
| 3 | 4 | 7 | 13.7% |
| 4 | 8 | 15 | 7.03% |
| 5 | 16 | 31 | 3.62% |
| 6 | 32 | 63 | 1.87% |
| 7 | 64 | 127 | 0.96% |
| 8 | 128 | 255 | 0.49% |
| 9 | 256 | 511 | 0.25% |
| 10 | 512 | 1,023 | 0.13% |
The catch: rare, but not rare enough
A single series almost always ends in a small win, which is what makes the system feel safe. But a series lasts only about 2 rounds on average, so a player goes through hundreds of them in an evening. Over 1,000 rounds, the chance of meeting the ten-loss streak is 47%, and when it comes it wipes out about a thousand earlier wins.
Why it cannot beat the house edge
An average series at 2x stakes about 11.5 base units in total and loses 0.34 units on average: exactly −3% of what was staked. The Martingale does not change the average. It reshapes the results into many small wins and one rare, very large loss.
Bankroll and table limits make it worse
To survive ten losses at 2x you need 1,023 times your base bet, all to win one base bet at a time. Real games also cap the maximum bet, which cuts the ladder short. When you cannot place the next doubling, the loss is locked in.
Martingale risk calculator
Calculator
At targets other than 2x the bet must grow faster or slower to recover losses: by target ÷ (target − 1) per step, which is ×3 at 1.5x and ×1.11 at 10x. The average cost stays 3% of everything staked.
Other progression systems
- Grand Martingale adds an extra unit on each doubling: faster growth, bigger ladder, same average.
- Reverse Martingale (Paroli) doubles after wins instead: rare big streaks, frequent small losses, same average.
- D’Alembert adds one unit after a loss and removes one after a win: slower swings, same average.
- Fibonacci follows the Fibonacci sequence after losses: a gentler Martingale, same average.
You can run the Martingale, D’Alembert and Paroli in Crash Lab and see the average land on −3% of the amount staked every time.
Questions
What if I stop after one win?
Stopping rules change which sessions end early; they do not change the average cost of the bets you place. Over many sessions the result is still −3% of the total staked.
What if I had unlimited money?
Nobody does, and every game caps the bet size. Even in theory, an unlimited bankroll would only be risking ever larger sums to win one base bet at a time.
Does it work better with a lower target such as 1.5x?
No. At 1.5x each loss must be followed by a bet three times larger, so the ladder grows much faster. The average stays the same.