D'Alembert

Another popular mathematical system is named after Jean Le Rond d'Alembert, a French mathematician and physicist who was born in 1717. His theory on the "Law of Equilibrium" supposes a balance of successes and failures of certain events if you consider a long series of these events. The d'Alembert, sometimes referred to as the "Pyramid System," has you increase your bet by one unit after a loss and decrease your bet by one unit after a win. The sequence and amount raised or lowered can be varied to suit particular games and odds.

Each loss go up one unit, each win go down one unit. Thus “hopefully” you get back where you started, meanwhile in the end making $1 from each and every win. Example:
1. First bet $1. If you win, bet $1. If you lose, bet $2.
2. If you win $2, next bet $1. If you lose, bet $3.
3. If you win $3, next bet $2. If you lose, bet $4.
4. If you win $4, next bet $3. If you lose, bet $5.
5. If you win $5, next bet $4. If you lose, bet $6.
6. If you win $6, next bet $5. If you lose, bet $8.
7. Etc.

You’d better set limits or will find yourself in outer space some day-but usually, D’Alembert is the most efficient system. For example if you start at $1 and “return to base” every 100 plays, whether winning or losing-this is definitely better than flat betting at $50.

One typical sequence would be handled as follows:
 

1 bet) Bet 1 unit	2 bet)Up to 2 units	3 bet) Bet 1 unit	4 bet) Up to 2 units	5 bet) Up to 3 units	6 bet) Bet 2 units
Lose,	Win,	Lose,	Lose,	Win,	Win,
-1 units	+1 unit	+0 units	-2 units	+1 unit	+3 units

Your "unit" can be equal to $1, $5, $25 or anything that you designate. If your unit were $5, then you would be down $5 after the first wager. Your second stake is $10 and the win puts you up to a net of one unit or $5. Now you decrease your next bet after a win, back to $5. The loss of $5 puts you even at zero units. The next bet of two units loses so you increase to three units. Because you win this wager, you will now decrease your stake to two units. This wager wins and you are up a total of three units thus far. There is no determined stop-win point with the system, so you must set one for yourself. If one unit profit were fine for you, then you would have won the sequence after the second wager (being up one unit) and quit or began a new sequence. If two or three units were your objective, then the sixth bet would have sufficed. The higher your objective win, the longer the sequence will be. You should also pre-select a stop-loss point for any sequence that you play to help control losses. Notice that this sequence has three wins and three losses. When the wins and losses balance each other, or are in equilibrium, then your net gain is equal to the number of wins in your sequence. This sequence has three wins that balance out three losses. The net gain is three units.

Please realize that if we had a losing sequence, a more aggressive unit size progression will work harder against you, losing money much faster. Because there are more ways to lose than win on an even-money wager (18 wins versus 20 losses out of 38 trials), you will be on the losing side of the sequence more often. Therefore, you may choose to portray a more favorable sequence here as an example. You are better, off in the end, losing less with the smaller unit size than winning more with a larger unit size. Let us examine something called a "tree diagram" of the d'Alembert system. For this example, we are using a $5 unit and will limit the progression to no more than five wagers:

The d´Alembert Tree Diagram (5-bet progression using a $5 unit)

Total Probabilities of sequence ending events = 1.00 or 100%

The tree diagram is called that because it spreads out as it grows, just as the possibilities do. Starting with one wager, you can easily see how all the possibilities develop going up to five bets deep. Once you know what all the possible outcomes are, you can calculate the likelihood of each terminal event on the tree. The terminal events are represented with rounded boxes and contain the probability of reaching that particular outcome. The chances of winning the first bet are easy to see. There are 18 ways out of 38 to win the wager; so, 18 divided by 38 equals 0.4737 or 47.37%. In order to win after the second bet you would have lost the first, then won the second. The chances of losing the first wager (20/38) times the chances of winning the second (18/38) are 24.93%. To calculate the probability of reaching a particular point on the tree diagram, just count the number of wins and loses along the way and apply them as exponents before multiplying everything together. We can calculate the likelihood of winning a sequence by losing three bets and winning two bets, for example, as in win #5:

P(Lose) x P(Lose) x P(Lose) x P(Win) x P(Win) = P(Win #5), which is the probability that this exact sequence will occur.

If: P(Win) = 18/38 and P(Lose) = 20/38, for each spin, then: (20/38)³ x (18/38)² = P(Win #5).

P(Win #5) = 0.0327 or 3.27%

If you calculate all the probabilities of terminal events and add them together, they should equal 1.00 (or 100%). A terminal event is an event that causes the progression to end. A situation where the bettor is ahead after the first through fourth bets would end the progression. After placing the fifth stake, win, lose or draw, we have decided to quit the sequence. Take the amount of money that we are ahead or behind for each terminal event and multiple it times the probability of that event. Now sum these up to calculate the average money won or lost for this particular betting system:
 

Win #1 ($5):	Win #2 ($5):	Win #3 ($5):	Win #4 ($5):	Win #5 ($5):
18/38 x $5	(20/38) x (18/38) x $5	(20/38)² x (18/38)² x $5	(20/38)³ x (18/38)² x $5	(20/38)³ x (18/38)² x $5
+$2.37	+$1.25	+$0.62	+$0.16	+$0.16
Average Total Winnings: +$4.56
Lose ($25):	Lose ($25):	Lose ($25):	Lose ($75):
(20/38)^4 x (18/38) x –$25	(20/38)^4 x (18/38) x –$25	(20/38)^4 x (18/38) x –$25	(20/38)^5 x –$75
–$0.91	–$0.91	–$0.91	–$3.02
Average Total Loses: <$5.75>

Allowing up to a 5-bet progression with $5 units, the D'Alembert delivers $4.56 in wins minus $5.75 in loses, for a net loss of $1.19 per betting sequence. Another useful bit of information is the average number of spins, or bets per progression. The summation of the number of spins times the probability of ending the progression in as many spins gives us this statistic. For the first four bets, the player must win to end the sequence. Otherwise, the sequence is automatically terminated after the fifth bet. You will note, there is no terminal event in the third spin, so the probability of ending the betting progression is zero. Here is how the calculation would look:

 

P(1 spin) x 1 spin	P(2 spins) x 2 spins	P(3 spins) x 3 spins	P(4 spins) x 4 spins	P(5 spins) x 5 spins
= P(Win #1), or 0.4737 x 1 spin	= P(Win #2), or 0.2493 x 2 spins	0 or 0.0 x 3 spins	= P(Win #3), or 0.0622 x 4 spins	= (1.0000 – .7852), or 0.2148 x 5 spins
= 0.4737	= 0.4986	= 0.0	= 0.2488	= 1.0740

Average number of spins for a 5-bet progression = 2.2951, or 2.3 spins.

We could have calculated the probability of all six terminal events in the fifth spin and added them together to get the probability of going to five spins. Because these events are mutually exclusive, it is easier take 1.00 minus the chances of ending the progression in spins one through four. The probability of ending in spins one through four is [0.4737 +0.2493 +0.0 +0.0622] or 0.7852. Therefore, we have 100% – 78.52%, which equals a 21.48% chance of ending the progression in the fifth spin. Taking the sum of all probabilities times the spins needed is about 2.3 average spins per progression for a 5-bet D'Alembert. If we lose $1.19 per progression and each progression averages 2.3 spins, then we are expecting a loss of almost 52 cents per bet.