# Martingale Betting System

## Introduction

There can be little debate that the Martingale is one of the oldest and most popular betting systems. The siren's call of the Martingale has fooled many gamblers as it usually results in a session win. However, everybody has a finite supply of money and eventually the Martingale will take down even the biggest of bankrolls.

## Rules

The following are the rules of the Martingale.

1. The Martingale is generally played on even money bets.
2. Start by deciding what your winning goal is and your bankroll size. The great the ratio of bankroll to winning goal, the greater your chances of a session success are.
3. To make the math easy, the ratio of bankroll to winning goal should be one less than some power of 2. For example, 7, 15, 31, 63, 127, 255, etc.
5. If the result of your last bet is a win, then you have achieved your winning goal and may quit.
6. If the result of your last bet is a loss, then double your next bet size, if you can.
7. If you can't double your bet because you're busted, then walk away broke.

In other words, keep doubling your previous bet, until you win. As long as you can keep doubling, you will achieve your winning goal eventually. No matter how many losses precede a win, you will achieve your winning goal exactly.

Let's look at an example of a player with a \$100 winning goal and a bankroll of \$6,300.

• His first bet is \$100, which loses.
• His second bet is \$200, which loses. He is now down \$300.
• His third bet is \$400, which loses. He is now down \$700.
• His fourth bet is \$800, which loses. He is now down \$1,500.
• His fourth bet is \$1,600, which loses. He is now down \$3,100.
• His fourth bet is \$3,200, which wins. He is now up \$100.

I don't think there is an official procedure if the player doesn't have enough money to double his last losing bet, but still has some money. I suppose what most players would do is bet it all. Then, keep betting the lesser of (1) whatever would be needed for achieve a session win and (2) everything left until either ending marker is achieved.

## What to Expect

The following tables show what you can expect to happen under different bets and bankroll sizes. Below is an explanation of each column.

• Maximum Bets — The number of losing bets you can afford to lose.
• Bankroll Units — This is the ratio of bankroll size to winning goal. It is equal to 2maximum bets - 1.
• Probability of Success — Probability of achieving winning goal.
• Average Wins per Loss — Number of units the player can expect to win between losses, assuming he keeps starting over after each session win or loss.
• Probability Doubling Bankroll — Probability of doubling the needed bankroll before going bust, assuming the player keeps starting over after each win.

The first table is based on the Player bet in baccarat. Not counting ties, the probability of the Player bet winning is 49.317517%.

### Baccarat — Player Bet

Maximum
Bets
Bankroll
Units
Probability
Success
Average Wins
per Loss
Probability
Doubling
Bankroll
3 7 0.869811 6.681152 0.376682
4 15 0.934017 14.155437 0.359188
5 31 0.966558 28.902712 0.348392
6 63 0.983051 58.000094 0.340634
7 127 0.991410 115.411214 0.334315
8 255 0.995646 228.687274 0.328694
9 511 0.997793 452.188677 0.323417
10 1023 0.998882 893.172207 0.318315

Let's look at the situation of the player being able to make up to six losing bets in a row. That would require a bankroll of 63 times the winning goal. The probability of achieving the winning goal of one unit is 98.3051%. If the player did this over and over, he could expect an average of 58.000094 winning sessions for each losing session of 63 units. Note that the loss of 63 units is greater than the 58 wins between losses. Finally, if the player keeps repeating this until he doubles his bankroll, one unit at a time, or goes broke, his probability of doubling his bankroll is 34.0634%.

The next table is based on the pass bet in craps. The probability of the pass bet winning is 244/495 = 49.292929%.

### Craps — Pass Bet

Maximum
Bets
Bankroll
Units
Probability
Success
Average Wins
per Loss
Probability
Doubling
Bankroll
3 7 0.869622 6.669984 0.376108
4 15 0.933889 14.126063 0.358450
5 31 0.966477 28.830284 0.347486
6 63 0.983001 57.828648 0.339557
7 127 0.991381 115.016656 0.333066
8 255 0.995629 227.797787 0.327272
9 511 0.997784 450.214759 0.321822
10 1023 0.998876 888.845840 0.316547

The next table is based on the don't pass bet in craps. The probability of the don't pass bet winning, not counting ties is 949/1925 = 49.298701%.

### Craps — Don't Pass Bet

Maximum
Bets
Bankroll
Units
Probability
Success
Average Wins
per Loss
Probability
Doubling
Bankroll
3 7 0.869666 6.672603 0.376243
4 15 0.933919 14.132952 0.358623
5 31 0.966496 28.847268 0.347699
6 63 0.983013 57.868843 0.339810
7 127 0.991387 115.109141 0.333359
8 255 0.995633 228.006247 0.327606
9 511 0.997786 450.677280 0.322197
10 1023 0.998877 889.859390 0.316962

The next table is based on any even money bet in single-zero roulette. The probability of an even money bet winning is 18/37 = 48.648649%.

### Single-Zero Roulette — Any Even Money Bet

Maximum
Bets
Bankroll
Units
Probability
Success
Average Wins
per Loss
Probability
Doubling
Bankroll
3 7 0.864588 6.384896 0.361132
4 15 0.930464 13.381113 0.339231
5 31 0.964293 27.005325 0.323946
6 63 0.981664 53.536685 0.311640
7 127 0.990584 105.203019 0.300746
8 255 0.995165 205.816405 0.290553
9 511 0.997517 401.747736 0.280730
10 1023 0.998725 783.298223 0.271123

The next table is based on any even money bet in double-zero roulette. The probability of an even money bet winning is 18/38 = 47.368421%.

### Double-Zero Roulette — Any Even Money Bet

Maximum
Bets
Bankroll
Units
Probability
Success
Average Wins
per Loss
Probability
Doubling
Bankroll
3 7 0.854206 5.859000 0.331848
4 15 0.923266 12.032100 0.301929
5 31 0.959614 23.760990 0.278607
6 63 0.978744 46.045881 0.258320
7 127 0.988813 88.387174 0.239598
8 255 0.994112 168.835630 0.221820
9 511 0.996901 321.687698 0.204736
10 1023 0.998369 612.106626 0.188263

## Video

Please enjoy my video on the Martingale.

## Summary

If you want to maximize your chances at a session win, then your chances of achieving it are best with the Martingale. This is at the expense of very short sessions (so less fun playing) and abject disaster when things go bad.

Like all betting systems, not only won't the Martingale eliminate the house edge, it won't even dent it. The expected ratio of losses to amount bet will approach the house edge the more you play. If you must use a betting system, please use a free one like the Martingale. Never pay for one, as they are all equally worthless mathematically.