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Last Updated: March 31, 2013

The Cancellation Betting System


Although I try to make it emphaticaly clear that all betting systems, including the cancellation system, are just gimmicks my viewers continue to ask me to recommend one. Any gambling systems can manipulate the probability of a winning session, at the cost of how much you lose for a losing session. In the long run there is one unavoidable truth, that the ratio of money lost to money bet will always approach a given constant. That constant depends on the game played but always equals the house advantage.

Not to condone gambling systems, but rather to prevent people from supporting those who scam the public by selling them, I shall present what is called the cancellation system here. It is also known as the Labouchere. This system is very similar to most other gambling systems in that you trade a few large losses for a lot of small wins. While some people like flying by the seat of their pants at the gambling table others prefer something more systematic. For them I am doing this.

How to Play

I learned about the "cancellation system" from viewer of this web page who believed strongly in it and asked for my opinion. I later saw a discussion of it in The Winner's Guide to Casino Gambling by Edwin Silberstang. The concept behind this system is that in the long run two events of roughly equal probability will happen roughly the same number of times. This system depends on this happening in the short run. The more money you wish to put up, the longer into the long run you can afford to go, thus increasing your odds of a winning session, but at the risk of losing more.

The goal of this system is to win a specified number of units at an even money game, like betting red or black in roulette or the pass or don't pass in craps. The player must decide how much a unit is and how many units to risk. The money risked for a particular session shall be called the “bankroll.” For purposes of example, we'll assume a ten unit win goal. Below are the specifics on how to use it:

  1. Start by writing on a piece of paper ten 1s in a row (1 1 1 1 1 1 1 1 1 1). Each number will represent a betting unit. When all numbers have been crossed off the paper you will have won 10 units.

  2. For each bet wager the sum of the number on the left and the number on the right, unless there is only one number left in which case you bet that number.

  3. If you win cross off both numbers.

  4. If you lose place the sum at the end on the right.

  5. If the bankroll you decided to risk at the beginning is not enough to bet the sum of the left and right numbers then:

    1. If your bankroll can cover the number on the left then wager that. If you win then cross off the left number. If you lose then add to the left number what you lost.

    2. If your bankroll can not cover the number on the left then bet whatever is left of your bankroll. If you win then deduct from the number on the left what you won. If you lose then your bankroll is consumed and you must leave the table.

  6. Keep repeating these steps until you either cross off all numbers or exhaust the bankroll you decided initially to risk.

Here is an example:

  • You start with 1 1 1 1 1 1 1 1 1 1.
  • You then wager two units and win, cross off the 1 on the left and right, leaving 1 1 1 1 1 1 1 1.
  • You then wager another two units and lose, adding a 2 to the right, leaving 1 1 1 1 1 1 1 1 2.
  • You then wager three units (1+2) and lose again, adding a 3 to the right, leaving 1 1 1 1 1 1 1 1 2 3.
  • You then wager four units (1+3) and win, crossing off the 1 on the left and the 3 on the right, leaving 1 1 1 1 1 1 1 2.

Here is an example of what to do if you are reaching the limit of your bankroll:

  • The numbers are 10 11 12 13 14 15. Your bankroll only has 20 units left.
  • Because you don’t have 25 units (10+15) you only bet the left number, 10, and win, leaving 11 12 13 14 15.
  • Your bankroll now has 30 units, enough to cover 26 (11+15), which you lose, leaving 11 12 13 14 15 26.
  • Your bankroll now only has 4, not enough to cover all of the left number so you bet all 4, and win, you then deduct 4 from the 11, leaving 7 12 13 14 15 26.
  • Your bankroll how has 8, so you bet the number of the left, 7, and lose, you then add what you lost to the left number, leaving 14 12 13 14 15 26.
  • Your bankroll now has only 1, you bet that and lose, leaving 15 12 13 14 16 26.
  • At his point your bankroll is gone and you must walk away, the total of all numbers left at this point will equal your the sum of 10 and your lost bankroll.

The theory is that by crossing off two numbers for a win and adding only one for a loss you will eventually cross off all the numbers. However in a bad session of losses the numbers start adding up fast and you can reach the end of your bankroll sooner than you might expect. The more you put up to risk the greater the probability of crossing off all the numbers. However the end result is always either winning ten units or losing the entire bankroll you put up to risk. Of course you are welcome to deviate from the strategy at any time. You can also start with more or less numbers, the more you put the more you can win but the greater the probability of losing your bankroll. The person who I learned this system from said that when he is nearing the limit of his bankroll he would chicken out and divide the numbers by two and play each set individually. While every gambling system I have ever seen says you must play their method exactly you are no better or worse off in the long run by deviating from any system. The discussion of this system in The Winner’s Guide to Casino Gambling states that you may use any numbers, but says the most common application is sequential numbers, for example 1 2 3 4 5. The purpose of gambling systems should be to increase your gambling enjoyment, not to change the outcome in the long run, so if you want to deviate then by all means you are free to do so.

Probability of Losing

To test this strategy I wrote a computer program to perform ten million sessions at various bankrolls on both roulette and craps. When applied to craps the bet was on the pass line which pays even money and has a probability of winning any given trial of 244/495 =~ 49.29%. In roulette the basic bet is any even money bet, which has a probability of winning of 47.37%. Naturally it is to the players advantage to apply this strategy, as well as any other, to craps as opposed to roulette. As with any betting system the ratio of the total amount lost to total amount bet was always very close to 1.41% in craps and 5.26% in roulette over the entire ten million sessions. Below are the results:

Results Against Roulette

Bankroll Probability of
Session Loss
Average Trials
per Session
Average Total
Bet per Session
5 72.9% 7.6 17.8
10 58.2% 11.7 31.1
20 42.2% 15.1 50.7
50 21.5% 19.3 90.0
100 15.3% 21.4 129.7
200 9.2% 22.8 177.2
300 6.8% 23.3 208.6
500 4.6% 23.8 252.8
750 3.3% 24.0 291.0
1000 2.7% 24.1 321.1

Results Against Craps

Bankroll Probability of
Session Loss
Average Trials
per Session
Average Total
Bet per Session
5 68.4% 7.8 18.3
10 52.2% 11.8 31.2
20 35.7% 14.8 49.1
50 18.6% 18.3 82.3
100 10.6% 19.9 112.6
200 5.7% 20.7 144.9
300 4.0% 21.0 164.8
500 2.5% 21.3 190.4
750 1.7% 21.4 211.6
1000 1.3% 21.5 227.0

The bankroll is in units, you decide how much a unit is. For example if you wish to risk $100 you might decide to make each unit $2 for a bankroll of 50. If you play this system against craps you have a 81.4% of winning 10 units, or $20, and a 18.6% chance of losing all 50 units, or $100.

Many systems out there are based on the kind of principle in this system. Instead of wasting your money by paying for a system you are welcome to use this one and use the money you saved to see a show. If you play this system, or any other, I wish you well but just don’t expect to show a profit over the long haul.

It should be noted that this system can be modified for any number of starting numbers, and they don't have to be ones. However, in the long run, no matter how the player uses it, he not only can't overcome the house edge, he can't even dent it.

Appendix

The computer program used to test this system can be found here. If you have a C++ compiler you are welcome to welcome to download it and make your own modifications if you desire. If you wish to run it but don’t have a compiler then if you ask nicely I may send it via e-mail.