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Mathematical Systems - Part 2

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There are no "good" mathematical betting systems in existence… only bad, very bad and worse yet, systems. Let me help redefine what a betting or mathematical system is.

Raise after a loss, raise after a win, or some convoluted method of tracking sequences of losses or wins to formulate subsequent wagers is a betting or mathematical system. These systems pay no attention what-so-ever to the physical conditions that caused the outcome! They do not even account for which game you are playing! No matter how great you've been told that your betting system is, or how fancy a name the peddler has attached to it, one thing is for certain. On games of independent trials, like roulette or craps, the odds of winning the next decision remain constant for the same bet. When you do win, you are paid incorrectly. American double-zero roulette wheels have 38 pockets, however the casino only pays you 35 to 1 (37 to 1 would be fair). This is how the house is able to stay ahead of the long-term player regardless of ANY betting system and run a thriving business! Please realize that a system based purely on whether or not you won or lost the last bet will wipe out your bankroll over a sustained period!

Because people will continue to use, or consider using betting systems, we will continue to discuss and compare various betting or mathematical systems. In Mathematical Systems Part 1, which is archived for your review, we talked about the Martingale and Grand Martingale systems in detail. Part 2 takes a look at the "d'Alembert" (or the Pyramid System) and the "Labouchere," (also called the Cancellation System).

The 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. His theory was misapplied to a betting system based on a much shorter span of casino outcomes. 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. One typical sequence would be handled as follows:

1) Bet 1 unit Lose, -1 units
2) Up to 2 units Win, +1 unit
3) Bet 1 unit Lose, +0 units
4) Up to 2 units Lose, -2 units
5) Up to 3 units Win, +1 unit
6) Bet 2 units Win, +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. I chose 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 dŽAlembert 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): 18/38 x $5 +$2.37
Win #2 ($5): (20/38) x (18/38) x $5 +$1.25
Win #3 ($5): (20/38)² x (18/38)² x $5 +$0.62
Win #4 ($5): (20/38)³ x (18/38)² x $5 +$0.16
Win #5 ($5): (20/38)³ x (18/38)² x $5 +$0.16
Average Total Winnings:   +$4.56
     
Lose ($25): (20/38)^4 x (18/38) x –$25 –$0.91
Lose ($25): (20/38)^4 x (18/38) x –$25 –$0.91
Lose ($25): (20/38)^4 x (18/38) x –$25 –$0.91
Lose ($75): (20/38)^5 x –$75 –$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(Win #1), or 0.4737 x 1 spin = 0.4737
P(2 spins) x 2 spins = P(Win #2), or 0.2493 x 2 spins = 0.4986
P(3 spins) x 3 spins 0 or 0.0 x 3 spins = 0.0
P(4 spins) x 4 spins = P(Win #3), or 0.0622 x 4 spins = 0.2488
P(5 spins) x 5 spins = (1.0000 – .7852), or 0.2148 x 5 spins = 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.

The Labouchere

The Labouchere system was named for a minister in Queen Victoria's service. Labouchere is credited for using the system although it is not known if he invented it. This betting method, also called the "Cancellation System," involves some record keeping. The player begins with a series of numbers – any series that he wishes to use. The series chosen will tally up to the number of units that the system player is trying to win. The player begins by betting the sum of the first and last number in the series. If the bettor wins this wager, he will cross out both of these numbers. If he loses, he will add the last bet made to the end of the series. Let's say, for example that the series used is 1-2-3-4-5-6. If the player is successful in canceling out the entire line, he will win exactly 21 units, or 1 +2 +3 +4 +5 +6. Let's play our example out to illustrate the mechanics of this system. The "x" denotes numbers that are canceled out after a winning bet:

Starting Line: 1-2-3-4-5-6 +0 units
1) Bet 1 +6, or 7 units and win: x-2-3-4-5-x +7 units
2) Bet 2 +5, or 7 units and lose: x-2-3-4-5-x-7 +0 units
3) Bet 2 +7, or 9 units and lose: x-2-3-4-5-x-7-9 -9 units
4) Bet 2 +9, or 11 units and win: x-x-3-4-5-x-7-x +2 units
5) Bet 3 +7, or 10 units and win: x-x-x-4-5-x-x-x +12 units
6) Bet 4 +5, or 9 units and lose: x-x-x-4-5-x-x-x-9 +3 units
7) Bet 4 +9, or 13 units and win: x-x-x-x-5-x-x-x-x +16 units
8) Bet 5 units and win: x-x-x-x-x-x-x-x-x +21 units

There are a couple of points of interest. First, you will see the 21 unit win, once the line is completely canceled out. Secondly, the bets start off high (7 units in this case) and can escalate quickly (up to 13 units) and this is a "friendly" sequence of wins and losses. Because the "5" is the only number remaining before the last wager, it represents the total bet to be made. If it had lost, then the next wager would have been this 5 plus the 5 added at the end to account for the loss. Each time a wins occurs, two numbers get canceled out, whereas each loss that you experience has you adding only one number to the end of the sequence. This is supposedly the selling point of the system. The series shrinks two numbers for a win, but only grows one number for a loss. The proponents forget to mention that the one number added is equal to the last bet which was the sum of two numbers. So you are trading one larger number for the removal of two smaller numbers. Let's look at a losing sequence to examine how quickly our wagers can mount:

Starting Line: 1-2-3-4-5-6 +0 units
1) Bet 1 +6, or 7 units and lose: 1-2-3-4-5-6-7 -7 units
2) Bet 1 +7, or 8 units and lose: 1-2-3-4-5-6-7-8 -15 units
3) Bet 1 +8, or 9 units and win: x-2-3-4-5-6-7-x -6 units
4) Bet 2 +7, or 9 units and lose: x-2-3-4-5-6-7-x-9 -15 units
5) Bet 2 +9, or 11 units and win: x-x-3-4-5-6-7-x-x -4 units
6) Bet 3 +7, or 10 units and lose: x-x-3-4-5-6-7-x-x-10 -14 units
7) Bet 3 +10, or 13 units and lose: x-x-3-4-5-6-7-x-x-10-13 -27 units
-Series is abandoned-

If you look at the following sequence, you will see that a couple of wins are sprinkled in for good measure, but the bettor never recovers from the first two losses. If our bettor were playing with $5 units, he would be down a whopping $135 with only three net losses (five losses minus two wins). Because there are limitless combinations of numbers to use and varying lengths of series available, it is impossible to analyze all the Labouchere series that can be created. For our average loss per spin calculations, which I've shown in great detail how to do for the d'Alembert, I have included three ultra conservative series of numbers, "1-2-1," "1-1" and the most conservative "1." The last of which is technically not a "series" of numbers. The 1-2-1 series will lose $5.60 per progression with 3.46 spins per progression. That works out to about $1.62 per spin when starting with $5 units. The less aggressive 1-1 will lose $1.99 per progression and last 2.5 spins, or about 80 cents a spin. The most conservative 1 series will average $1.14 per progression for 2.3 spins which is about 50 cents a spin. Even with a win objective of only four units per sequence (as with the 1-2-1) this system can be crippling to your bankroll. Like other systems, the more money wagered, or exposed to the house edge, the more money lost on average.

You must realize that a longer series of larger numbers has a remote likelihood of success. The shorter the series and smaller the values, the better the chance of winning the series. Of course, we are minimizing our exposure, and hence limiting our losses. Be careful with the Labouchere. It is one of the worst systems you can employ (right up there with the Grand Martingale).

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