Explainer · 9 min read
d'Alembert and Labouchère: two betting systems worked through, and why neither beats the margin
The d'Alembert and Labouchère systems promise steadier progress than doubling up. One adds a unit after each loss; the other keeps a list of numbers to cross off. Both are clever bookkeeping. Neither can change what a single bet is worth, and that is the part that decides your result.
Updated 22 October 2026
Contents
How the d'Alembert system works
Named after the 18th-century French mathematician Jean le Rond d'Alembert, this system is meant for bets at odds of about 2.00. You choose a unit, say €10, and:
- after a loss, add one unit to the next stake;
- after a win, take one unit off (never going below one unit).
The thinking is that wins and losses should even out, and because winning stakes are on average a little larger than losing ones, you end up ahead. Here is a run of twelve bets with six wins and six losses:
| Bet | Result | Stake | Running total |
|---|---|---|---|
| 1 | Lost | €10 | −€10 |
| 2 | Lost | €20 | −€30 |
| 3 | Won | €30 | €0 |
| 4 | Lost | €20 | −€20 |
| 5 | Won | €30 | +€10 |
| 6 | Won | €20 | +€30 |
| 7 | Lost | €10 | +€20 |
| 8 | Lost | €20 | €0 |
| 9 | Lost | €30 | −€30 |
| 10 | Won | €40 | +€10 |
| 11 | Won | €30 | +€40 |
| 12 | Won | €20 | +€60 |
Six wins, six losses, +€60. That is the system on a good day. The weakness is the assumption that wins and losses even out. At real bookmaker prices they don't quite: odds of 2.00 usually carry a true chance of about 48%, so losses slightly outnumber wins over time, and each surplus loss pushes the stake up by another unit. After eight straight losses you have lost €360 and the next stake is €90. That sounds mild, but the stake level rarely comes back down during a long bad stretch.
How the Labouchère system works
Labouchère, also called the cancellation system, uses a written line of numbers. Their sum is your target profit in units. With the line 1-2-3-4 and a unit of €10, the target is €100.
- Your stake is always the first plus the last number on the line.
- Win: cross off both numbers.
- Loss: write the amount you just lost at the end of the line.
- When the line is empty, you have reached the target. Start a new line.
| Bet | Line before | Stake | Result | Running total | Line after |
|---|---|---|---|---|---|
| 1 | 1-2-3-4 | €50 | Lost | −€50 | 1-2-3-4-5 |
| 2 | 1-2-3-4-5 | €60 | Won | +€10 | 2-3-4 |
| 3 | 2-3-4 | €60 | Lost | −€50 | 2-3-4-6 |
| 4 | 2-3-4-6 | €80 | Lost | −€130 | 2-3-4-6-8 |
| 5 | 2-3-4-6-8 | €100 | Won | −€30 | 3-4-6 |
| 6 | 3-4-6 | €90 | Lost | −€120 | 3-4-6-9 |
| 7 | 3-4-6-9 | €120 | Won | €0 | 4-6 |
| 8 | 4-6 | €100 | Won | +€100 | (empty) |
Four wins, four losses, target reached. The seductive part: each win removes two numbers and each loss adds only one, so in theory you need to win only a little over a third of your bets to clear a line. A handy check: the numbers left on the line always add up to the target minus your current result.
Labouchère in a bad run
The same rules after a weaker stretch (three wins in nine bets):
| Bet | Line before | Stake | Result | Running total |
|---|---|---|---|---|
| 1 | 1-2-3-4 | €50 | Won | +€50 |
| 2 | 2-3 | €50 | Lost | €0 |
| 3 | 2-3-5 | €70 | Lost | −€70 |
| 4 | 2-3-5-7 | €90 | Lost | −€160 |
| 5 | 2-3-5-7-9 | €110 | Lost | −€270 |
| 6 | 2-3-5-7-9-11 | €130 | Won | −€140 |
| 7 | 3-5-7-9 | €120 | Lost | −€260 |
| 8 | 3-5-7-9-12 | €150 | Lost | −€410 |
| 9 | 3-5-7-9-12-15 | €180 | Won | −€230 |
The line is now 5-7-9-12, which adds up to €330: the €100 target plus the €230 you are down. The next stake is €170, from a €50 start, to chase a €100 goal. Every loss makes the line longer and the end numbers bigger, so stakes grow quickly even though you are winning a third of the time.
Why no progression changes expected value
Here is the short proof, without heavy maths. Take bets at odds of 2.00 with a true win chance of 48%. Each euro staked has an expected value of 0.48 × 2.00 − 1 = −4%.
your total result = result of bet 1 + result of bet 2 + … + result of bet n
expected result of bet i = stake i × (−4%)
expected total = (stake 1 + stake 2 + … + stake n) × (−4%)
The key point is that every progression sets the next stake before the next match is played, using only past results. The match doesn't know your history, so its win chance is still 48% whatever your stake. A sum of bets that are each negative on average is negative on average. No arrangement of stakes, however clever, adds a positive term. Mathematicians formalise this as the optional stopping theorem: with limited money and time, no betting strategy turns a losing game into a winning one.
What progressions do change is the shape of the results: how often you win small, how often you lose big. That is variance, not value.
Bankroll and stake limits
We simulated 20,000 runs of 1,000 bets with a €1,000 bankroll and a €10 unit at odds of 2.00, counting how often the system asked for a stake larger than the money left:
| System | Win chance 48% | Win chance 50% (no margin) |
|---|---|---|
| d'Alembert (+1 unit after a loss, −1 after a win) | about 97% of runs | about 82% |
| Labouchère (line 1-2-3-4, restarted after each target) | about 99% of runs | about 95% |
Even at a perfectly fair 50%, the average ending balance was about €1,000 in both systems, exactly where it started. With a normal bookmaker margin, it was well below. On top of that, bookmakers set maximum stakes, and accounts that bet in obvious patterns may be limited. When the system calls for a stake you can't place, the series is simply lost. Raising stakes to win back losses is chasing, a recognised warning sign; our responsible gambling page lists free help.
All progression systems compared
| System | After a loss | After a win | Typical pattern | Next stake after 8 losses (€10 base) | EV per € at 48% |
|---|---|---|---|---|---|
| Martingale | Double | Back to base | Many small wins, rare huge loss | €2,560 | −4% |
| Fibonacci | One step up | Two steps down | Slower climb, long recoveries | €340 | −4% |
| d'Alembert | +1 unit | −1 unit | Gentle, but drifts upwards while losing | €90 | −4% |
| Labouchère | Add stake to the line | Cross off two numbers | Line and stakes balloon in bad runs | €130 (line 1-2-3-4) | −4% |
| Paroli | Back to base | Double (up to 3 wins) | Many small losses, rare bigger win | €10 | −4% |
| Flat staking | Same stake | Same stake | Steady, swings follow the odds | €10 | −4% |
The last column never moves. Only the quality of the bets can change it. That is why approaches that work on the bets themselves, such as sure betting, where every outcome is covered across bookmakers, are a different category from staking systems. They carry their own risks (odds changes, voided bets, account limits), but they don't rely on a progression. Our overview of betting strategies compared puts them side by side.
Frequently asked questions
Does the d'Alembert system work for sports betting?
No. It wins when wins and losses balance out, but bookmaker odds carry a margin, so losses slightly outnumber wins over time and the stake keeps rising during bad stretches. The expected result is the total amount staked times the negative edge in the odds.
Why does Labouchère seem to need only a third of bets to win?
Each win crosses off two numbers and each loss adds only one, so a line can be cleared while losing most bets. But every loss also makes the stakes bigger, so the money at risk grows fast and the bankroll or a maximum stake usually ends the line first.
Can any staking progression beat the bookmaker margin?
No. Each stake is set before the match from past results, and the match outcome doesn't depend on them. The expected result of every bet stays negative at normal odds, and a sum of negative expectations is negative.
Which betting progression is the least risky?
Flat staking keeps swings smallest because the stake never grows after losses. Paroli limits each cycle's loss to one base stake. Neither changes the expected value: at a 48 percent win chance and odds of 2.00, every system loses about 4 percent of the money staked on average.
Sources
- Wikipedia: Betting system (d'Alembert and other progressions)
- Wikipedia: Labouchère system
- Wikipedia: Optional stopping theorem
- Wikipedia: Gambler's ruin
- Wikipedia: Expected value
- Spielsuchthilfe (Austria): counselling for gambling problems
As of October 2026. Simulation by us: 20,000 runs of 1,000 independent bets each; all figures are worked examples, rounded.
Betting involves risk, including sure bets: prices move, bets get voided and accounts get limited. Only bet money you can do without. Our responsible gambling page lists free, confidential help.
Not betting, financial, legal or tax advice. 18+. As of October 2026.