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D' Alembert progression for roulette

Jean Le Rond d’Alembert (1717-1783) was a French mathematician, physicist and and music theorist. D’Alembert’s formula for obtaining solutions to the wave equation is named after him.

D’Alembert progression is a money management type of roulette system that can be found in many flavors, all having in common that you add and subtract a fixed amount to your bet, after a loss or a win. This can be done in a negative way (up-as-you-lose) and in a positive (up-as-you-win), it can have different sizes of the amounts to be added and subtracted and there can be stop-loss instructions. The initial bet can be different as well, depending on the author.

Jean Le Rond d'Alembert roulette system

The Concept

This is mainly an even chances system for Red-Black, High-low, Even-Odd. Let’s have a look at the basic idea. Your initial bet is 5 units and after a loss you add one unit and after a win you subtract one from the latest bet. You Bet 5, Win 5, Bet 4 (you subtract one because you won), Lose 4, Bet 5 (raise one unit as you lost), Lose 5, Bet 6 (raise again as you lost again), Win 6. Because you won, you lower the bet to 5 units…

What happened here? We had two losses (4 and 5 units) and two wins (5 and 6 units). Because we add one unit at a loss and subtract one at a win, this will give as a result that any loss will be smaller than the previous or next win; we bet 5 and won and then bet 4 and lost. The loss is one unit less than the previous win and so we have a profit of one unit and we are back to our initial bet. Then we bet 5 and lost, we added one unit, bet 6 and won. The lost bet was one unit less than the next won. This means, that any pair Win/Lose (or Lose/Win) will bring you one unit in profit – no matter the ORDER of wins and losses: Bet 5, Lose 5 (Total: -5), Bet 6, Lose 6 (Total: -11), B 7, L 7 (-18), B 8, Win 8 (-10), B 7, W 7 (-3), B 6, W 6 (+3), and we’re back at our initial bet of 5 units. As you see here, we had three losses (5, 6, 7) and three winnings (8, 7, 6) in the order L-L-L-W-W-W. The three pairs of W/L are first bet L5 / last bet W6 (+1), L6/W7 (+1) and L7/W8 (+1) making the positive net of three units. Try any combination of three wins and three losses and you will have the same end result.

But this is not all! As we, in this case, had an initial bet of 5 units, it can be lowered in five steps and if you win that last bet (1 unit) you have gained an ADDITIONAL profit of 15 units (5 + 4 + 3 + 2 + 1)! This will happen every time you have five wins MORE than losses.

It is ALSO possible to have a positive net result when there are MORE LOSSES than wins, if the number of pairs (one unit profit each) are less than the added UNITS of all “un-matched” losses. Maybe you have, after spin 22, ten wins and twelve losses. You can then “match” ten pairs, giving you a net gain of 10 units. If the two additional losses’ added VALUES are less than that you still have a profit, if they are more you have a loss.

So, then! If every pair Win/Lose will give us a profit, as well as any excess winnings, and we have a “safety net” for some excess losses if we get that; how on Earth can this system lose?

If we use this system in, for example, a coin-flipping game where the odds are exactly 50/50 for the two outcomes to show (yes, I know it can stand on its edge – or a bird can snatch it while it’s in the air, but…), then you will have no problems. As soon as the winnings are equal to the losings – in numbers – you will have earned one unit for every two decisions.

Another example

In roulette, however, there is nothing like a 50/50 chance as the chance for the ball to land on the zero or 00 is 1/37 and 2/38 resp (way more than the chance of a coin landing on its edge – or being stolen by a bird – I think). This means that you will face more losses than wins (in the long term) giving you ever increasing bets. And just as I described above, those “un-matched” losses must be payed for from the one-unit profits you have gained from your winning bets. Let me show:

Bet 5, Lose 5 (-5), B 6, L 6 (-11), B 7, W 7 (-4), B6, L 6 (-10), B 7, L 7 (-17), B 8, W 8 (-9), B 7, W 7 (-2)

Let’s look at this. We have had three winnings and four losings in this series and we need another hit to make it to our initial bet; the next bet is 6 and if we win, we are down to 5 again and a new series begins. But we know as a fact that the losings will outnumber the winnings, at least at a pace of one in 37 (or two! in 38 for a 00-wheel – boycott them). So if this was your “1/37 extra loss” (supposing there is such a thing), you will need two wins to reset only (and a gaurantee that the rest of the bets will be equally won and lost). But if the rest of the spins are equally won and lost, you will be stuck here at the higher-than-the-initial bet as you need MORE wins than losses, from now on, to get down to the initial 5 units.

Modifications

Can there be something done to change all this, then? Many a good man has tried. None, that I know of, have had success… But I will show you some examples:

One very common way to try to solve the problem is to lower the bet TWO (or more!) steps after a win, in a try to “drift” towards zero in a higher rate than towards the table limit. This will also mean that less winnings are required to keep the bet on a steadily decreasing amount – two hits in five spins, to be exact. Yes, you need only two winnings in five spins – not 50%… It’s TRUE, but that’s a sales-person’s argument. Let me re-phrase that: “You need only two winnings in five spins to keep the bet amount decreasing – whether you like it or not!”. That’s it, you don’t like it as that will result in a lot of “un-matched” losses when you are back to your original bet and so you are FORCED to gamble the 5 – 4 – 3 – 2 – 1 bets – supposing you started at 5…

Positive progression (up-as-you-win) can also be shown. The idea is that a loss, as it will come more often than a win, lowers the bet towards zero while winning will raise the bet until some limit is hit; the table limit for example.

In this situation we have the exact opposite of the above: for every pair Win/Lose we have, we LOSE exactly one unit as the losing bet will always be higher than the winning previous or next bet. Here is a sequence:

B 5, W 5 (+5), B 6, W 6 (+11), B 7, L 7 (+4), B 6, L 6 (-2) and we’re back to 5…

Two pairs W/L and we are two units down. And should we want to really risk our money, we can bet the 5 – 4 – 3 – 2 – 1 levels too, losing another 15 units. The idea here is not to balance with the losses; it’s to find streaks of winners so that your bet increases and if so, you’ll make loads of money. The only problem is, of course, to select the winners…

Lowering the bet TWO (or more!) units on a loss will put us, again, in a situation opposite to the scheme described above: For only TWO losses in five spins the bet will decrease, AND it will give us one “un-matched” WINNING – is this a safety-net? No, unfortunately not as the back side is that MORE than two losses in five spins (as will most of the time be the case) will outnumber the winnings and that is as bad for this “back-words” way as for the normal way. The fact is; you will need more winnings than losings for this system to work!

The d’Alembert progression/money management plan or any variation of it is not recommended for actual gambling. Although it is a very wide-spread and well known roulette betting method it is considered not a winning system and we have included it in our “failing systems” list. 

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