How the D’Alembert system works in roulette

The D’Alembert system is a stake progression: start with one base unit, add one unit after a full loss, and subtract one unit after a win, never going below one unit. It changes your stakes and bankroll exposure—not the probability of winning. Profit is not guaranteed, and the order of wins and losses affects the result.

All calculations below are conditional models, not forecasts. The worked examples assume an even-money bet with a 1:1 net payout: a win earns the stake amount as profit, while a full loss costs the entire stake. A 1:1 payout does not mean a 50% chance of winning.

The stake rule and base unit

Set the next stake from the previous result: add one base unit after a full loss or subtract one after a win, with a minimum stake of one unit.

The unit is the base stake amount you choose. Your bankroll is the money set aside for play, not a guarantee that a session is safe.

With a $10 unit:

• Start with a $10 stake.
• After losing $10 in full, stake $20.
• After losing $20 in full, stake $30.
• After winning with a $30 stake, reduce the next stake to $20.
• After winning with a $10 stake, keep the next stake at $10.

This rule covers full losses. If zero causes only a partial loss, you need to define separately how that settlement changes the next stake.

A worked sequence ending $60 ahead

With a $500 bankroll and a $10 base unit, the fixed sequence L,L,W,L,W,W,L,W,W,W ends with a $560 balance—a net profit of $60 on this hypothetical path.

The table shows the stake before the spin and the balance after settlement. It assumes 1:1 net winnings, full losses, sufficient funds for each listed stake and no table limit. All amounts are whole dollars, so no rounding is needed.

This is the calculator’s hardcoded example, not a randomly generated sample or evidence that the system is profitable.

D’Alembert sequence table with a 500-dollar bankroll and a 10-dollar base unit English interface captured from localhost, not production. At the 390px mobile width, the table scrolls horizontally: only the first columns are visible, and the remaining columns are off-screen.

Why the order of results matters

The same number of wins and losses can produce different results because the sequence determines the stake on each spin.

Starting at one unit u, with 1:1 net winnings and full losses:

• LLWW: −u − 2u + 3u + 2u = +2u.
• WWLL: +u + u − u − 2u = −u.

With a $10 unit, the paths look like this:

Both sequences contain two wins and two losses. One finishes two units ahead; the other finishes one unit behind. The one-unit floor contributes to the difference: a win at the minimum stake cannot reduce the next stake any further.

Cumulative losses and the money needed to continue

After k consecutive full losses, cumulative losses are u × k(k+1)/2, and the next stake is (k+1)u. These formulas assume a start at one unit, no table limit and no settlement changes.

The loss formula adds the stakes u + 2u + 3u and so on through ku. Funding those losses is not the same as having enough left to place the next stake.

For a $500 bankroll and a $10 unit:

• Nine consecutive full losses use stakes from $10 through $90.
• The cumulative loss is $450.
• The remaining bankroll is $50.
• The next required stake is $100, which cannot be funded in full.

So the bankroll funds nine consecutive full losses, but not the next stake. The balance is not zero: $50 remains. This is an inability to continue the prescribed progression, not literal bankruptcy.

Exposure after five losses: flat betting, D’Alembert and Martingale

After five consecutive full losses, flat betting loses 5 units, the D’Alembert system loses 15 units, and Martingale loses 31 units.

This comparison is a conditional model of one path. It assumes the same $10 starting unit, 1:1 net winnings, sufficient bankroll and no table limit. Flat betting keeps the stake unchanged; D’Alembert adds one unit after a loss; Martingale doubles the stake after a loss.

These figures describe exposure along that losing path. They are not a universal ranking of the systems or estimates of their probability of profit.

Roulette probabilities, zero settlement and house edge

Under the standard European single-zero model, a red or black bet wins on 18 pockets and loses in full on 19 of the 37 pockets. Zero counts as a full loss in this model.

The expected net result per unit staked is (18−19)/37 = −1/37, giving a 2.7027% house edge.

Under the standard double-zero model, both zero and double zero cause full losses: there are 18 winning pockets and 20 losing pockets out of 38. The expected net result per unit is (18−20)/38 = −1/19, giving a 5.2632% house edge.

These figures apply only to the stated wheel and settlement models. Half-loss zero rules change the figure. In New Jersey, Casino Gaming Regulations, section 13:69F-5.2, allow an outside even-money bet on a double-zero roulette table to lose half or all of the stake when 0 or 00 lands, at the licensee’s option with notice to players. This is not a universal rule: check your table’s rules, as zero settlement varies by table and jurisdiction. The 5.2632% figure above assumes a full loss on both zeros. If both zeros instead result in a half-loss, the 38-pocket model—with 18 wins, 18 full losses and two half-loss outcomes—gives a house edge of 1/38, or 2.6316%.

In the conditional model of independent spins, previous results do not change the next spin’s probability. Starting from one fixed point, the probability of k consecutive full losses in the standard European model is (19/37)^k.

The displayed percentages are rounded to four decimal places. They are not session-wide risk-of-ruin probabilities or guarantees. They describe one run with a specified start, not every possible losing run during a session, the ability to fund future stakes or the final balance.

How to use the embedded calculator

Enter your bankroll, base unit and target profit to inspect the maximum funded consecutive-loss count and the fixed example sequence. The calculator does not forecast future roulette results.

To reproduce the article’s example:

1. Enter 500 in the bankroll field.
2. Enter 10 in the base unit field.
3. Enter 100 in the target profit field.
4. Compare the displayed stakes and balances with the worked table above.

Local D’Alembert calculator form showing a 500-dollar bankroll, 10-dollar base unit and 100-dollar profit target English interface captured from localhost, not production: bankroll $500, base unit $10 and target profit $100.

The calculator’s “bets needed” figure is a hardcoded heuristic without a probability model. Its formula is ceil(target/unit × 4): divide the target by the unit, multiply by four and round up to a whole number. It has no model of the outcome path.

Do not use that figure as a forecast of how many bets will achieve the target, as a probability estimate or as evidence of profitability. Entering a target does not make it achievable, and the calculation does not make your chosen base unit “optimal.”

How to run the simulator and read the comparison

Set the simulator’s inputs and select Run Simulation. The output is a random illustration under your selected assumptions—not a prediction or a calibrated risk-of-ruin measure.

The simulator provides these inputs:

1. Bankroll.
2. Base unit.
3. Win probability.
4. Bet count.
5. Simulation count.

Choose the win probability according to the wheel and settlement model, not the payout alone. An even-money payout does not justify entering a 50% win probability.

The simulator uses unseeded Math.random, so repeated runs with identical settings can produce different sequences and balances. It treats every non-win as a full loss and does not model a table limit.

It is not a test of roulette fairness or an RNG validation tool. Its full-loss model also cannot represent a table with partial-loss zero settlement without changing the assumptions.

The strategy comparison below also uses a random, unseeded path. It illustrates the results of a particular run; it cannot establish a universal ranking of betting systems.

For a separate planning exercise, you can open the staking-plan calculator. Its Kelly context and risk label are not roulette outcome estimates or session-wide risk-of-ruin estimates.

What the calculation tells you

The D’Alembert system lets you calculate the next stake and the cost of a specified consecutive losing streak. It cannot predict outcomes or guarantee profit.

Check three things separately: the order of results, the funds available for the next stake and the table’s zero-settlement rules. In the $500 bankroll example with a $10 unit, nine full losses leave $50—not a zero balance, but too little to continue the progression.