D’Alembert Betting System

A description of the d’Alembert Betting System, with examples of basic play and how to apply it in various situations that come up at the table.

The d’Alembert Betting System

Many of the progressive betting systems used in casino table games today were actually developed by some of the greatest scientific minds of the 18th century. French mathematicians, in particular, were fascinated with theories of probability, especially as they related to tossing coins, rolling dice, and other activities of chance. Among them was Jean-Baptiste le Rond d’Alembert (1717~1783), who was also a noted physicist, philosopher, and music theorist.

Despite his vast knowledge of math and physics, d’Alembert incorrectly reasoned that the probability of a tossed coin coming up heads would increase for every time that it landed tails. He referred to this as the “Law of Equilibrium,” having observed that when two events are equally likely to occur, they really do, in the long term, appear to occur equally. Therefore, the basic strategy behind d’Alembert’s system of betting was to decrease one’s bet when winning and to increase it the more one loses.

In truth, however, all coin flips are independent outcomes, unaffected by past results. Many gamblers labor under the same mistaken belief, that a streak of losses must be counterbalanced by a streak of wins, that each player will get as many natural blackjacks as the dealer does, and so on. It is not at all surprising that so many casino table game players gravitate toward progressive betting systems such as the one named for d’Alembert and based upon his ideas of equilibrium.

Basic Play

The d’Alembert’s betting system is quite easy to implement at the Blackjack table. It assumes that the number of hands a player wins and loses will, over the long term, result in a ratio that is very close to 50:50. Compared to many other popular progressions, such as Martingale and Labouchere, d’Alembert involves no calculation more difficult that being able to add or subtract one unit from the bet previously made.

The objective is to win a single unit in profit, and the player begins by wagering one unit. Each time a bet wins, one unit will be subtracted from the total wagered for the next bet. Each time a bet loses, one unit will be added to the total wagered for the next bet. It is just that simple.

For example, if $5 is the basic unit and the first bet wins, subtracting one unit results in a next wager of zero, so the progression ends with a profit of $5. Then, the progression can begin again.

On the other hand, following a loss, the wager would be increased by one unit to $10. If it wins win, reduce the wager by one unit back to $5. If it loses, increase it one unit to $15. Continue playing in this way until the required wager is zero. A single unit in profit will have been won. Then, start the progression again.

Applying the System

Unlike the Martingale Betting System, d’Alembert does not require risking huge amounts at unfavorable odds to recover previous wagers lost. And unlike Labouchere, even a string of losses will never increase the wager by more than a single unit on the subsequent bet. It is a very slow and relatively less risky progression than either of the other two.

Because the system was developed for coin flips and other games of chance where only an even-money outcome is possible, its application to Blackjack cannot be entirely straightforward. That’s because splitting, doubling down, insurance, and higher odds paid for a natural 21 make Blackjack a bit more complicated than even-money bets on Red or Black at Roulette and Pass or Don’t Pass at the Craps table.

When splitting pairs or doubling down, the additional wagers cannot be treated as part of the ongoing progression, so a strategy for handling them is required. Wins are certainly no cause for concern, as the additional income can be treated as a bonus. This is also true when a blackjack hand pays 3-to-2 odds.

Similarly, a push can be ignored; the progression can go on by re-betting the previous amount and continuing the series. And the insurance option can simply be ignored, as if it were not even part of the game—which is actually a good strategy in any event.

A real challenge to the progression only arises when the double-down loses or both of the split hands fail. In this case, the recommended course of action is to treat the additional amount lost as a separate account, to be paid for out of the “bonuses” won, as above. This maintains the integrity of the progression for the next hand, and it is congruous with the theory of equilibrium—wins and losses on additional wagers should be about equal in the long term.


  • Trevor Hallsey

    Passionate webmaster, devoted card game enthusiast, and proud son of the Great White North. With over a decade of iGaming experience, Trevor has launched numerous web portals to share his passion for game theory and all things Canadian gaming. With this site, he acts as a fact checker and mostly writes at the intersection of gaming and finance. He aims to offer statistical insights and unique information that you might see lacking in similar sites.

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