The question
"If I double up after every loss, the first win gets it all back. How can that lose?"
It's the oldest idea in gambling, and it's been sold as a secret for centuries. This lesson answers "does the martingale betting system work?" with the maths and a simulation of 200,000 players, and shows why Fibonacci, d'Alembert and every other progression fail for the same reason.
The idea in one sentence
Changing how much you stake can't change the expected value of each bet, so a staking system with no edge underneath only reshapes your losses: many small wins, then one catastrophe.
The picture
Martingale feels safe because it almost always works. Each cycle ends with a small win, and you have to lose many times in a row to fail. But the stakes double every time you lose, so the one cycle that fails wipes out dozens that succeeded.
We simulated 200,000 players each placing up to 1,000 bets at 2.0 on a true 50% chance (fair odds, so the 2% commission makes each bet −1%). Everyone starts with £1,000 and a base unit of £10. A player stops when they can't afford the next stake the system calls for.
| System | Finished in profit | Couldn't place the next stake | Average result | Median result | Average turnover | Loss as % of turnover |
|---|---|---|---|---|---|---|
| Level £10 stakes | 36.5% | 0.4% | −£99 | −£100 | £9,995 | −0.99% |
| Martingale | 18.8% | 94.4% | −£85 | −£470 | £8,238 | −1.03% |
| Fibonacci | 22.8% | 77.2% | −£117 | −£604 | £11,618 | −1.00% |
| d'Alembert | 14.0% | 85.9% | −£280 | −£884 | £27,910 | −1.00% |
Look at the last column. Every system lost 1% of what it staked: exactly the expected value of one bet. The systems only changed how often you win, how big the disasters are, and how much you stake along the way.
Over a short session the martingale looks brilliant. Over just 100 bets, half the martingale players finished in profit (median +£113), and the other half ran into six straight losses and couldn't continue.
| System | In profit | Stopped | Mean P&L | Median P&L | Mean turnover | Loss as % of turnover |
|---|---|---|---|---|---|---|
| Martingale | 19.2% | 93.8% | −£49.70 | −£470.20 | £8,442 | -0.59% |
| Loss no. | Stake | Total lost |
|---|---|---|
| 1 | £10.20 | £10.20 |
| 2 | £20.62 | £30.82 |
| 3 | £41.65 | £72.47 |
| 4 | £84.16 | £156.63 |
| 5 | £170.03 | £326.66 |
| 6 | £343.53 | £670.20 |
| 7 | £694.08 | can't afford it |
Worked Betfair example
Over/Under 2.5 Goals in a match where the over is a genuine 50% chance, priced fairly at 2.0. Bank £1,000. Goal: £10 profit per cycle, after 2% commission. (Illustrative figures.)
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What a win pays. Each £1 staked at 2.0 wins £1.00, less 2%: £0.98.
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First stake. To win £10 you need 10 ÷ 0.98 = £10.20.
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After each loss, stake enough to recover everything plus £10. Stake = (losses so far + £10) ÷ 0.98:
Loss number Next stake Total lost if it loses too 1st bet £10.20 £10.20 2nd bet £20.62 £30.82 3rd bet £41.65 £72.47 4th bet £84.16 £156.63 5th bet £170.03 £326.66 6th bet £343.53 £670.20 7th bet £694.08 can't afford: only £329.80 left -
So one cycle is a bet on "not six losses in a row". The chance of six straight losses at 50% is 0.5⁶ = 1 in 64, or 1.56%.
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Expected value of one cycle. 63/64 × £10 − 1/64 × £670.20 ≈ 9.84 − 10.47 = −£0.63. Every cycle loses 63p on average, even though 98.4% of them win.
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How it plays out. On average you'd expect about 63 winning cycles (+£630) before the first bust (−£670.20). The median player in our simulation won 20 cycles (+£200), then hit the bust (−£670.20), finishing on −£470.20.
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What if you had a bigger bank? A £10,000 bank survives about three more losses in a row, so it busts less often, but when it does the loss is about eight times bigger. The expected loss per £ staked doesn't move from −1%.
Verdict: martingale doesn't beat the maths. It turns a steady 1% cost into a small, frequent profit and a rare, huge loss, and the rare loss always comes.
Why Fibonacci and d'Alembert fail too
- Fibonacci (after a loss move one step up 1, 1, 2, 3, 5, 8…; after a win move two steps back) rises more gently than doubling, so it survives longer but still hits a stake it can't afford.
- d'Alembert (one unit up after a loss, one down after a win) rises slowly, so it rarely busts quickly, but it stakes far more in total. In our table its turnover was nearly three times level staking, so it lost nearly three times as much.
- Every "recovery" plan assumes a win is due after losses. It isn't: each bet is independent (Lesson 3.5).
The formula
The expected value of any staking plan
- EV per £1 is the expected profit of one bet per £1 staked. Here 0.5 × 0.98 − 0.5 = −0.01.
- E[total staked] is the average turnover the system produces.
In plain English: each bet loses 1p per £1 on average, whatever the size of the stake and whatever happened before it. Add them up and the total expected loss is 1% of everything you staked. Systems that stake more lose more.
The martingale cycle
- q is the chance of losing one bet (0.5 here).
- k is how many losses in a row your bank can survive (6 here).
- T is the target profit per cycle (£10).
- L_k is the total lost after k straight losses (£670.20).
In plain English: a cycle is a big chance of a small win against a small chance of a huge loss. With no edge, the huge loss always outweighs the small wins on average.
With a genuine edge?
If each bet has a positive EV, then more turnover means more expected profit, and staking plans do change your result, but it's the edge doing the work. Chasing losses with an edge still raises your risk of ruin enormously. Proportional staking such as fractional Kelly is the sensible way to size a real edge.
Try it
Choose martingale, odds 2.0, 50%, £10 base and a £1,000 bank, and check the bust point is six losses in a row. Then run 1,000 bets for each system and compare the "loss as % of turnover" column: it lands near −1% every time.
Common mistakes
- "It always wins in the end." It wins most sessions. Over enough sessions, the losing one arrives, and it's bigger than all the wins before it.
- Thinking a bigger bank fixes it. It makes the disaster rarer and bigger. The expected loss per £ staked is unchanged.
- Believing a win is "due". A 50% chance is 50% after six losses, just as it was before them (Lesson 3.5).
- Judging a system on a few weeks of results. Martingale's results are hugely skewed. A short, happy record says almost nothing about the risk you're carrying.
- Looking for the answer in staking instead of in the price. Profit comes from getting better odds than the true chance. Staking only decides how bumpy the ride is.
Why an edge, not a system, is the only thing that makes money: why good models still lose money.