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Last Updated: May 12, 2020

# Labouchere Betting System

## Introduction

The Labouchere (also known as the cancellation and split-Martingale) is a well-known betting system. It is generally played on even-money games. It entails writing a list of numbers and then crossing off two after a win and adding one after a loss. Like most betting systems, it usually results in a session win, but at the expense of huge losses when it doesn't.

The logic behind the Labouchere is that the player needs to only win 1/3 of the time (and not run out of money) for the system to work. Believers might say the Law of Large Numbers states that the more bets are made, the more the ratio of wins to bets will approach the expected ratio. While this is true, a bad string of losses can causes bets to increase quickly, sometimes exhausting the player's bankroll.

## Rules

1. The player will define his winning goal and bankroll.
2. The player will divide his winning goal by 10.
3. The player will write down the quotient from step 2 ten times, horizontally.
4. The player will bet the sum of the left and right numbers in the list on an even money bet*.
5. If the player wins the bet in step 4, then he will cross off the left and right numbers on the list.
6. If the player loses the bet in step 4, then he will add a new element to the right end of the list consisting of the amount just lost.
7. Go back to step 4, unless every element has been crossed off the list (success) or the player's bankroll is exhausted (failure)

*: Exceptions

• If there is only one element in the list, bet that.
• If the player's bankroll is greater than zero, but not enough to cover the the sum of left and right elements in the list, then bet the remaining bankroll. If this bet wins and the bet amount is less than the element on the right end of the list, then subtract the amount bet from the right element. If this bet wins and is greater than the element on the right end of the list, then cross off the element on the right side and deduct the amount bet less the right element from the left element.

## Example

• In this example, the player's goal is to win \$50 and his bankroll is \$1,000. He starts by dividing \$50 by 10 and writing the quotient ten times: 5-5-5-5-5-5-5-5-5-5.
• The player's first bet is \$10 (\$5 + \$5), which he wins, and crosses off the right and left elements: 5-5-5-5-5-5-5-5.
• The player's second bet is \$10 (\$5 + \$5), which he wins, and crosses off the right and left elements: 5-5-5-5-5-5.
• The player's third bet is \$10 (\$5 + \$5), which he wins, and crosses off the right and left elements: 5-5-5-5.
• The player's fourth bet is \$10 (\$5 + \$5), which he loses, and adds to the right end of the list, leaving: 5-5-5-5-10.
• The player's fifth bet is \$15 (\$5 + \$10), which he loses, and adds to the right end of the list, leaving: 5-5-5-5-10-15.
• The player's sixth bet is \$20 (\$5 + \$15), which he wins, and crosses off the right and left elements: 5-5-5-10.
• The player's seventh bet is \$15 (\$5 + \$10), which he loses, and adds to the right end of the list, leaving: 5-5-5-10-15.
• The player's eight bet is \$20 (\$5 + \$15), which he loses, and adds to the right end of the list, leaving: 5-5-5-10-15-20.
• The player's ninth bet is \$25 (\$5 + \$20), which he loses, and adds to the right end of the list, leaving: 5-5-5-10-15-20-25.
• The player's tenth bet is \$30 (\$5 + \$25), which he wins, and crosses off the right and left elements: 5-5-10-15-20.
• The player's 11th bet is \$25 (\$5 + \$20), which he wins, and crosses off the right and left elements: 5-10-15.
• The player's 12th bet is \$20 (\$5 + \$15), which he loses, and adds to the right end of the list, leaving: 5-10-15-20.
• The player's 13th bet is \$25 (\$5 + \$20), which he loses, and adds to the right end of the list, leaving: 5-10-15-20-25.
• The player's 14th bet is \$30 (\$5 + \$25), which he wins, crossing off the left and right elements, leaving: 10-15-20.
• The player's 15th bet is \$30 (\$10 + \$20), which he wins, crossing off the left and right elements, leaving: 15.
• The player's 16th bet is \$15 (the only element left), which he loses, adding to the right end of the list: 15-15.
• The player's 17th bet is \$30 (\$15+\$15), which he loses, adding to the right end of the list: 15-15-30.
• The player's 18th bet is \$45 (\$15+\$30), which he loses, adding to the right end of the list: 15-15-30-45.
• The player's 19th bet is \$60 (\$15+\$45), which he loses, adding to the right end of the list: 15-15-30-45-60.
• The player's 20th bet is \$75 (\$15 + \$60), which he wins, crossing off the left and right elements, leaving: 15-30-45.
• The player's 21st bet is \$60 (\$15 + \$45), which he wins, crossing off the left and right elements, leaving: 30.
• The player's 22nd bet is \$30 (the only element left), which he wins, declaring victory and walks out with \$50 in winnings.

## Simulation Results

To show what to expect from using the Labouchere, I wrote a simulation that followed the rules above, based on various bets and games. The simulation used a Mersenne Twister random number generator. For each simulation, the winning goal was ten units. I tested the simulation on the following bankrolls: 50, 100, 250, 500, 1000, and 2500 units.

The first simulation is based on betting the Player bet in baccarat. The simulation size is over 6 billion sessions. As a reminder, the theoretical house edge on the Player bet is 1.235%.

### Baccarat Simulation — Player Bet

Statistic 50 Units 100 Units 250 Units 500 Units 1000 Units 2500 Units
Probability winning goal reached 81.47% 89.52% 95.34% 97.53% 98.70% 99.45%
Average number of bets 17.810 19.565 20.766 21.184 21.405 21.542
Average units bet 90.651 123.859 171.423 209.457 249.284 304.325
Expected win per session -1.119 -1.528 -2.113 -2.588 -3.084 -3.765
Ratio money lost to Money bet 1.234% 1.234% 1.233% 1.236% 1.237% 1.237%

The next simulation is on the pass bet in craps. As a reminder, the house edge on the pass bet is 7/495 = 1.411%. The simulation size is 8.7 billion sessions.

### Craps Simulation — Pass Bet

Statistic 50 Units 100 Units 250 Units 500 Units 1000 Units 2500 Units
Probality winning goal reached 81.40% 89.47% 95.31% 97.51% 98.69% 99.45%
Average number bet 17.821 19.583 20.791 21.213 21.435 21.573
Average total bet 82.123 112.285 155.551 190.203 226.522 276.844
Expected win per session -1.160 -1.586 -2.198 -2.689 -3.201 -3.915
Ratio money lost to Money bet 1.413% 1.413% 1.413% 1.414% 1.413% 1.414%

The next simulation is based on the don't pass bet in craps. The simulation size was over 6 billion sessions. As a reminder, the house edge on the don't pass bet is 1.364%.

### Craps Simulation — Don't Pass

Statistic 50 Units 100 Units 250 Units 500 Units 1000 Units 2500 Units
Probality winning goal reached 81.41% 89.48% 95.31% 97.52% 98.70% 99.45%
Average number bet 17.819 19.579 20.786 21.206 21.427 21.566
Average total bet 84.447 115.440 159.904 195.465 232.766 284.361
Expected win per session -1.152 -1.571 -2.182 -2.666 -3.171 -3.863
Ratio money lost to Money bet 1.364% 1.361% 1.365% 1.364% 1.362% 1.358%

The next simulation is based on any even money bet in single-zero roulette. The simulation size was over 6.9 billion sessions. The theoretical house edge in single-zero roulette is 1/37 = 2.703%.

### Roulette Simulation — Single Zero

Statistic 50 Units 100 Units 250 Units 500 Units 1000 Units 2500 Units
Probality winning goal reached 79.52% 88.01% 94.41% 96.93% 98.33% 99.26%
Average number bet 18.123 20.070 21.465 21.978 22.257 22.441
Average total bet 84.738 117.939 167.599 209.142 254.264 319.542
Expected win per session -2.291 -3.188 -4.527 -5.652 -6.877 -8.648
Ratio money lost to Money bet 2.703% 2.703% 2.701% 2.703% 2.705% 2.706%

The next simulation is based on any even money bet in double-zero roulette. The simulation size was over 8 billion sessions.

### Roulette Simulation — Double Zero

Statistic 50 Units 100 Units 250 Units 500 Units 1000 Units 2500 Units
Probality winning goal reached 75.45% 84.71% 92.23% 95.44% 97.34% 98.71%
Average number bet 18.697 21.046 22.881 23.615 24.046 24.351
Average total bet 89.839 129.507 193.860 252.141 320.010 426.256
Expected win per session -4.729 -6.817 -10.204 -13.270 -16.843 -22.427
Ratio money lost to Money bet 5.263% 5.264% 5.264% 5.263% 5.263% 5.261%

## Video

Please enjoy my video explaining the Labouchere and what you can expect if you play it.