What is D’Alembert? Starting with the basics
Roulette consists of a wheel with numbered pockets and a ball that lands in one of them. A European wheel has 37 pockets: 18 red, 18 black and one green zero. Each time the ball is released and produces a result is called a spin.
A bet on red depends on the ball landing in a red pocket. Under the rules used here, a win returns the amount placed and adds a profit of the same amount. If the result is black or zero, the entire amount is lost.
For example, if two units are placed on red and red wins, four units are returned: two are the original amount and the other two are profit. The actual gain is two units, not four.
A unit is simply a measure used to keep track of the figures. It does not represent a recommended amount of money.
D’Alembert is not a way of finding out where the ball will land. It is a rule for changing the amount after each result. That is why it helps to separate two questions: how much is at stake and what the chances of winning are. The system changes the first, but not the second.
How the example in this article works
This article uses a specific version of D’Alembert:
- The starting amount is one unit.
- After a loss, the next amount increases by one unit.
- After a win, the next amount decreases by one unit.
- The amount never falls below one unit.
After losing one unit, the model therefore moves to two. Another loss takes it to three. If that spin wins, the next amount falls to two.
This means that a win does not automatically return the system to its starting amount. If the amount was five units, a win would reduce it to four, even if the overall balance was still negative.
All the examples assume a balanced European wheel, with an equal chance of landing in each pocket and spins that are independent of previous results. Red remains the favourable result; black and zero are treated as full losses.
We also specify how many spins each example covers. We do not assume that the person can carry on indefinitely or must continue until losses have been recovered. Some versions of roulette treat zero differently, so their figures would be different.
What happens after several losses and one win
Imagine that the model produces three losses followed by a win.
The first spin loses one unit. The second loses another two. The third loses another three. At that point, the losses add up to six units.
The fourth amount is four units, and this time the result is favourable. That spin produces a profit of four, but six had already been lost.
| Spin | Amount wagered | Outcome | Profit or loss | Running balance |
|---|---|---|---|---|
| 1 | 1 | Loss | -1 | -1 |
| 2 | 2 | Loss | -2 | -3 |
| 3 | 3 | Loss | -3 | -6 |
| 4 | 4 | Win | 4 | -2 |
The win reduces the accumulated loss, but does not remove it. The balance is still two units below the starting point.
Although eight units are returned on that fourth spin, four are the return of the amount placed. Only the other four are profit.
Under the rule defined above, the next amount would be three units. The system moves down one step after a win; it does not check whether earlier losses have been recovered.
This example shows why “there has been a win” and “the money has been recovered” are different statements.
Why as many wins as losses does not guarantee a profit
Some D’Alembert examples end in profit and can look convincing.
Suppose the model first loses one unit, then two, and then wins with three and with two. The losses total three and the profits total five. The final balance is two units ahead.
There have been two wins and two losses, but the wins occurred at larger amounts.
Now consider another fictional example.
The first five spins are all wins. Because the minimum amount is one unit, each win adds one and the accumulated profit reaches five.
Five losses follow. The amounts are one, two, three, four and five units. Those losses total fifteen.
The combined result is a loss of ten units: five were gained and fifteen were then lost.
Here, too, there are as many wins as losses. Yet the result is negative.
The difference lies in the order and the amounts associated with each result. Counting wins is not enough to establish the balance. Choosing only a sequence that ends well does not demonstrate that the system will produce a profit when the results change.
How losses build up even when the amounts rise slowly
Moving from one unit to two, then three and then four may look like a small increase. But each new loss is added to all the earlier ones.
This table shows six consecutive losses:
| Consecutive loss | Amount lost on that spin | Total lost since the start |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 3 |
| 3 | 3 | 6 |
| 4 | 4 | 10 |
| 5 | 5 | 15 |
| 6 | 6 | 21 |
The final amount is six units, but the total lost has already reached twenty-one. Looking only at the amount on the last spin hides much of what has happened.
After those six losses, the next step in the model would be seven units. Another seven would need to be available, in addition to the twenty-one already lost.
The amounts grow more slowly than when they are doubled. That describes a difference between systems, but does not make the accumulated losses small or guarantee their recovery.
Nor does it take an endless losing streak to encounter difficulties: a limited sequence can use up the available money.
What happens when money runs out or a table limit intervenes
A rule can specify the next amount without making it possible to place it.
Imagine a fictional starting balance of twenty units. After five consecutive losses of one, two, three, four and five, fifteen have been lost. Five remain.
The next D’Alembert step would require six units. With only five left, the defined rule cannot be followed. The balance does not have to reach zero for the model to be interrupted.
There is a separate restriction: the maximum amount allowed by the table.
Suppose that maximum were five units per bet. After a loss at five, the next step would be six and would not be permitted, even if enough money remained.
These examples are not a way to find a “suitable” balance or limit. They show that any explanation assuming unrestricted continuation leaves out an important part of the problem.
Even if the next spin could be played, there is no guarantee that it would win or that its profit would make up for what had been lost.
Why the wheel does not make up for previous losses
After several losses, an understandable feeling can arise: a win should be due by now.
But the wheel has no outstanding debt. Under the assumptions of the example, there are still 18 red pockets and 19 that are not red: the 18 black pockets and zero.
The chance of red remains 18 out of 37, approximately 48.65%. It does not increase because several black results or zeros have appeared before it.
Zero also explains why red and black do not give an exact 50% chance for these bets. The profit on a win equals the amount placed, but there is one more unfavourable pocket than favourable ones.
With these payouts and a full loss on zero, the casino’s mathematical advantage is approximately 2.70% of the total amount wagered.
This is not a fee deducted on every spin, nor does it mean every session loses exactly that percentage. Some sessions end in profit; others produce larger or smaller losses. The figure describes the mathematical average, not a guaranteed individual outcome.
The total amount wagered is also different from the starting money: the same money can be put at stake more than once.
Adding one unit after a loss changes the money that depends on the next result. It does not remove zero or improve the payout offered.
How it differs from Martingale and Paroli
All three systems change amounts, but they respond to results differently.
In the classic Martingale system, the amount doubles after a loss. Consecutive losses produce amounts of one, two, four, eight units and so on.
In D’Alembert, the increase after a loss is one unit. With the same losing sequence, the amounts would be one, two, three and four. After a win, the amount decreases by one unit, subject to the defined minimum.
In the Paroli example examined in the other article, amounts increase after wins. It considers a sequence of one, two and four units, ending at the first loss or after three wins.
These differences change how much can be won or lost in particular sequences. They do not change the pockets on the wheel or make a favourable spin more likely.
Slower increases or a stopping rule do not demonstrate that a model is profitable. Understanding its risks also requires looking at unfavourable cases, accumulated amounts and restrictions that can prevent it from continuing.
What recording a session can teach us
Some moments are easier to remember than others. A win after several losses may be remembered as a recovery, even when the figures are still negative.
Recording what happened helps separate that impression from the results. In the first example, there was a four-unit win, but the balance was still minus two. In the ten-spin example, there were as many wins as losses, yet ten units were lost.
It also helps distinguish an observed streak from a prediction. Knowing that a colour has appeared several times describes the past; it does not say what must happen next.
RouletteLab allows results to be recorded and descriptive statistics for a session to be viewed. It can help show frequencies and streaks and compare impressions with the recorded numbers. It does not execute strategies or calculate the profits or losses in these examples.
The value of examining D’Alembert lies in understanding how a simple rule can produce very different results depending on the sequence. Amounts rise a little at a time, but losses add up. None of those changes alters the chances on the next spin or removes the house edge.