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Last Updated: May 28, 2020

# Fibonacci Betting System

## Introduction

The Fibonacci betting system is method of betting that usually ensures a session win, at the expense of large losses when things go bad. The system uses the Fibonacci numbers to press bets after a loss. Much like the Labochere, as long as the player wins at least 1/3 of even-money bets, which usually have a 48% to 49.5% chance of winning, then the Fibonacci will result in a session win, as long as the player doesn't run out of money trying.

## Fibonacci Numbers

Fibonacci Numbers are a sequence of numbers that appear all over mathematics and nature. One could write a whole book about it. For purposes of understanding the betting system, one just needs to know the sequence.

The first two numbers in the sequence are 1 and 1. After that, each subsequent number is the sum of the previous two numbers. Another way of stating it is if F(n) is the nth Fibonacci Number, then:

• F(1) = 1
• F(2) = 1
• F(n) = F(n-1) + F(n-2), where n>=3

The first 25 Fibonacci numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025.

That said, I believe most other web sites incorrectly explain how to use the Fibonacci. The way they explain it, many sessions would result in a session push. Nobody uses a betting system to just push. The way I explain it always results in a win of the initial bet.

Here is how to play the Fibonacci:

1. Decide on your winning goal and establish a bankroll* you are willing to risk to achieve that goal. Call your winning goal g.
2. At any given time using this system, you will be somewhere on the Fibonacci sequence. You will start on the SECOND one**.
3. Bet the product of g and your place in the Fibonacci sequence.
4. If the result of step 3 is a win, then go back two places on the Fibonacci sequence. If this causes you to fall off the sequence, then walk away a winner.
5. If the result of step 3 is a loss, then go up one place on the Fibonacci sequence.
6. If you still have money, then go back to step 3.

Footnotes:

*: Ideally, the bankroll should be the sum of consecutive Fibonacci numbers, starting with the second one. The first 25 of such sums are: 1, 3, 6, 11, 19, 32, 53, 87, 142, 231, 375, 608, 985, 1595, 2582, 4179, 6763, 10944, 17709, 28655, 46366, 75023, 121391, 196416.
**: This is where I believe most other sources on the Fibonacci are wrong, which say to start with the first 1 in the sequence. About half the time, this will result in a session push, for example if the first two outcomes are a loss and a win.

## Example

The player decides his winning goal is \$10.

### Example

Bet
Number
Bet
Amount
Win/Loss Balance
after Bet
1 \$ 10 L \$ -10
2 \$ 20 L \$ -30
3 \$ 30 L \$ -60
4 \$ 50 W \$ -10
5 \$ 20 L \$ -30
6 \$ 30 W \$ 0
7 \$ 10 L \$ -10
8 \$ 20 L \$ -30
9 \$ 30 L \$ -60
10 \$ 50 L \$ -110
11 \$ 80 L \$ -190
12 \$ 130 W \$ -60
13 \$ 50 W \$ -10
14 \$ 20 L \$ -30
15 \$ 30 W \$ 0
16 \$ 10 W \$ 10

Note that despite winning 6 bets out of 16 only (37.5%), the player finally comes out ahead. Also note he had to wager up to \$130 to achieve that. Had the player lost that bet, he would have been down \$320. A lot to risk losing for a small win.

## Simulation Results

The first simulation is based on betting the Player bet in baccarat. The top row shows the size of the bankroll in units. The simulation size is over 37 billion sessions. As a reminder, the theoretical house edge on the Player bet is 1.235%.

### Baccarat Simulation — Player Bet

Statistic 11 Units 19 Units 32 Units 53 Units 87 Units 142 Units
Probability winning goal reached 0.911630 0.946062 0.966720 0.979286 0.987048 0.991879
Average number of bets 2.615806 2.927595 3.151269 3.318705 3.438306 3.524126
Average units bet 4.897283 6.383979 7.948692 9.593549 11.303999 13.074258
Expected win per session -0.060439 -0.078760 -0.098253 -0.118545 -0.139764 -0.161316
Ratio money won to money bet -0.012341 -0.012337 -0.012361 -0.012357 -0.012364 -0.012338

The next simulation is based on betting the Pass bet in craps. The top row shows the size of the bankroll in units. The simulation size is over 100 billion sessions. As a reminder, the theoretical house edge on the Pass bet is 1.414%.

### Craps Simulation — Pass Bet

Statistic 11 Units 19 Units 32 Units 53 Units 87 Units 142 Units
Probability winning goal reached 0.911445 0.945909 0.966610 0.979203 0.986989 0.991834
Average number of bets 2.368097 2.650841 2.853825 3.005671 3.114277 3.192349
Average units bet 4.434771 5.783194 7.203172 8.696173 10.249543 11.859770
Expected win per session -0.062661 -0.081814 -0.101862 -0.123059 -0.144991 -0.167709
Ratio money won to money bet -0.014130 -0.014147 -0.014141 -0.014151 -0.014146 -0.014141

The next simulation is based on betting the Don't Pass bet in craps. The top row shows the size of the bankroll in units. The simulation size is over 104 billion sessions. As a reminder, the theoretical house edge on the Don't Pass bet is 1.364%.

### Craps Simulation — Don't Pass Bet

Statistic 11 Units 19 Units 32 Units 53 Units 87 Units 142 Units
Probability winning goal reached 0.911486 0.945945 0.966636 0.979222 0.987004 0.991845
Average number of bets 2.435480 2.726184 2.934764 3.090872 3.202506 3.282736
Average units bet 4.560666 5.946911 7.406284 8.941055 10.537201 12.192076
Expected win per session -0.062165 -0.081108 -0.101012 -0.122033 -0.143677 -0.166225
Ratio money won to Money bet -0.013631 -0.013639 -0.013639 -0.013649 -0.013635 -0.013634

The next simulation is based on betting any even money bet in double-zero roulette. The top row shows the size of the bankroll in units. The simulation size is over 97 billion sessions. As a reminder, the theoretical house edge on even money bets in double-zero roulette is 5.263%.

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

Statistic 11 Units 19 Units 32 Units 53 Units 87 Units 142 Units
Probability winning goal reached 0.895965 0.933393 0.956825 0.971718 0.981368 0.987675
Average number of bets 2.464700 2.794335 3.042827 3.237881 3.384956 3.496308
Average units bet 4.720054 6.309737 8.068833 10.016054 12.152640 14.489878
Expected win per session -0.248416 -0.332131 -0.424767 -0.527211 -0.639619 -0.762458
Ratio money won to Money bet -0.052630 -0.052638 -0.052643 -0.052637 -0.052632 -0.052620

The next simulation is based on betting any even money bet in single-zero roulette. The top row shows the size of the bankroll in units. The simulation size is over 88 billion sessions. As a reminder, the theoretical house edge on even money bets in double-zero roulette is 2.703%.

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

Statistic 11 Units 19 Units 32 Units 53 Units 87 Units 142 Units
Probability winning goal reached 0.906470 0.941950 0.963570 0.976918 0.985304 0.990611
Average number of bets 2.400123 2.698082 2.915528 3.080911 3.201387 3.289492
Average units bet 4.528809 5.954938 7.482040 9.116027 10.847380 12.675059
Expected win per session -0.122356 -0.160996 -0.202195 -0.246408 -0.293289 -0.342601
Ratio money won to Money bet -0.027017 -0.027036 -0.027024 -0.027030 -0.027038 -0.027030

## Video

Please enjoy my video on the Fibonacci betting system.