Jean le Rond d’Alembert was one of the sharpest minds of 18th-century Paris, co-editor of the Encyclopédie and the author of serious work on mechanics. He also believed, wrongly, that after a run of heads a coin was more likely to land tails. Gamblers took that one error, attached his name to it, and built a staking plan around it. The D’Alembert system assumes that results balance out, so it bets a little more after each loss and a little less after each win.
Its progression is the gentlest of the classic systems, which makes it the most comfortable to play. It does not make it cheaper. The edge on a single-zero wheel stays at 2.70% of every dollar staked, as it does for each plan covered in our roulette strategy hub. This page gives the exact formula for what the system pays, so you can see where the balance idea breaks.
The rules
Pick a starting stake and a unit; the classic version uses the same amount for both, say C$5. Bet on red, black, odd, even, 1-18 or 19-36.
- After a loss, raise the stake by one unit.
- After a win, lower it by one unit, but never below the starting stake.
That is all. No sequence to remember, no line to cross out, and a stake that drifts up and down by small steps.
A 12-spin example in Canadian dollars
Here is a session at C$5 start and C$5 unit, with six wins and six losses:
Swipe the table
| Spin | Result | Stake | Running total |
|---|---|---|---|
| 1 | Loss | C$5 | minus C$5 |
| 2 | Loss | C$10 | minus C$15 |
| 3 | Win | C$15 | level |
| 4 | Loss | C$10 | minus C$10 |
| 5 | Win | C$15 | plus C$5 |
| 6 | Win | C$10 | plus C$15 |
| 7 | Loss | C$5 | plus C$10 |
| 8 | Loss | C$10 | level |
| 9 | Loss | C$15 | minus C$15 |
| 10 | Win | C$20 | plus C$5 |
| 11 | Win | C$15 | plus C$20 |
| 12 | Win | C$10 | plus C$30 |
Six wins, six losses, C$30 profit. Each win landed on a stake one unit higher than the loss it answered, so the session banked one unit per win. This is the result that sells the system. If wheels produced equal wins and losses, the D’Alembert would print money.
The same session put C$140 through the wheel. Its expected result, before any spin, was C$140 ÷ 37, a loss of about C$3.78. This run was simply luckier than average.
The closed-form result
You do not need to simulate the D’Alembert to know where it ends up. If the stake never drops to its floor along the way, the outcome depends on just four numbers: the starting stake S, the unit u, the number of wins W and the final gap D between losses and wins (D = losses minus wins).
Result = u × W − S × D − u × D × (D − 1) ÷ 2
Check it on the example: S = C$5, u = C$5, W = 6, D = 0. The result is C$5 × 6 = C$30, matching the table.
Read the formula slowly, because it holds the whole story. The first term adds one unit for every win. The second term charges you a starting stake for each loss you are behind. The third term grows with the square of the gap. When losses lead wins by 2, it costs one unit; by 6, it costs 15 units; by 10, it costs 45 units. Small imbalances are cheap. Large ones get expensive quickly, and that is where the losses come from.
What 40 spins look like
Take a slightly roomier version: start at C$25 with a C$5 unit, which gives the stake space to fall before it touches the C$5 floor. Over 40 even-money spins on a European wheel, here is what the formula gives for the most common splits, with the chance of each split:
Swipe the table
| Wins | Losses | Gap D | Result | Chance of this split |
|---|---|---|---|---|
| 16 | 24 | 8 | minus C$260 | 6.99% |
| 17 | 23 | 6 | minus C$140 | 9.35% |
| 18 | 22 | 4 | minus C$40 | 11.32% |
| 19 | 21 | 2 | plus C$40 | 12.42% |
| 20 | 20 | 0 | plus C$100 | 12.36% |
| 21 | 19 | minus 2 | plus C$140 | 11.15% |
(The results assume the stake never hits its floor during the session. Sessions where wins run far ahead touch the floor and come out a little different.)
The pattern is lopsided. Being two losses behind still leaves you up. Four behind is a small loss. Eight behind costs C$260, more than double what twenty-all earns. Because an even-money bet wins only 18 times in 37, the gap drifts in the wrong direction on average, and the larger gaps carry the heavier weight. We ran 200,000 such sessions through a simulation with the floor in place: the average session lost C$33.39, even though about 60% finished ahead, a share that falls as sessions get longer. The average amount staked was C$1,227.76, and 2.70% of that is C$33.18. Simulation noise accounts for the gap of 21 cents.
The classic C$5 version over 100 spins
The textbook D’Alembert, C$5 start and C$5 unit, tells the same story over a longer session. We simulated 200,000 sessions of 100 even-money spins on a European wheel. The average session lost about C$90, close to the C$88 predicted from 2.70% of the average C$3,273 staked. About 58.6% still ended ahead, down from 60% over 40 spins, and the median session finished C$65 up, which is exactly the kind of result that keeps the system popular. One session in twenty lost more than C$955, and one in a hundred more than C$1,840.
That spread is the D’Alembert in a sentence: the median evening shows a small gain, the average evening loses the edge on everything staked, and the bad evenings are large enough to pay for all the good ones and the house edge besides.
Why the balance idea fails
The D’Alembert rests on the same mistake its namesake made: that the wheel tends toward balance. It does not. The proportion of wins does creep toward 18/37 over thousands of spins, but the raw gap between losses and wins does not shrink. It tends to grow, roughly with the square root of the number of spins, plus a steady drift of one spin in 37 toward the house. The formula punishes exactly that growth.
There is also a plainer problem. Each spin is its own even-money bet, priced at minus 1/37 of whatever is on it. The system decides the stake. It cannot decide the odds. A roulette odds calculator shows that minus 2.70% for any single stake you try, and summing those figures over the session gives the expected result directly.
Long sessions and the table limit
Over short sessions the D’Alembert feels safe, and in a narrow sense it is: the stakes climb by C$5 a time, and seven straight losses from a C$5 start cost only C$140. Over long sessions the gap has room to grow. In our simulation of 200-spin sessions from the same C$25 start, the average loss rose to about C$295 and the share ending ahead dropped to just under 53%. The worst 1% of sessions lost more than C$4,000.
A table maximum puts a lid on the stake, which makes things worse once you are deep behind: the stake can no longer grow enough for the closing wins to repair the gap. At a C$500 cap with a C$5 unit, the stake hits the ceiling when losses lead wins by about 95, which a typical evening never reaches. The budget usually runs out long before the table limit does.
The contrarian version
Some players reverse the rule: up one unit after a win, down one after a loss. This contrarian D’Alembert likes streaks and dislikes choppy tables, the mirror image of the original. It is a matter of taste. Its expected cost per dollar staked is the same 2.70% on a single-zero wheel and 5.26% on a double-zero wheel.
Playing it sensibly
If you enjoy the slow rhythm, the D’Alembert is a reasonable way to pace a session, provided the money is a spending budget and nothing more. Set the session amount and a time limit before you start, and stop at whichever arrives first. Never raise the unit to close a gap; the formula above shows what a larger gap costs at a larger unit. Our bankroll and session limits guide has worked budgets, and the responsible gambling page lists the limit tools Canadian sites offer.
Choose the wheel with the same care. Single-zero tables are standard at the sites listed on our Ontario roulette page, and a French table with La Partage cuts the cost of even-money bets to 1.35%. For the record, d’Alembert never recommended any such betting plan himself. He argued about coins in an essay titled Croix ou pile, and the system’s inventors seem to have read only the title.