The Fibonacci numbers are defined as:    

So for example if we assume that and then we get :  

(1)

Now if we divide the Fibonacci number in this period by the Fibonacci numbers in the previous period an 

interesting pattern called the Golden Mean is emerging 

Now the Fibonacci betting strategy is a type of Martingale betting system where the player is  

increasing his bet size when he is losing. The bet size at any point in time for a losing sequence is given by 

which again are equal to the Fibonacci numbers.  

The Fibonacci betting strategy can be illustrated as seen below.  

The bet size for the Fibonacci betting strategy in a losing sequence is increasing much slow compared

to a Martingale betting strategy. 

We should also note that the amount of money required as collateral is different from the bet size at any point. 

At any point in time we need to be able to cover our previous bets which means that we need to have at least 

the amount of money seen in the chart below in order to pursue a Fibonacci betting strategy .   

We now assume that we bet on the outcome of a random 0,1 coin toss. If the outcome of the coin toss is 0

then we lose and if the outcome of the coin toss is 1 then we win. 

	(2)

We can see that in this particular case with only 30 period our total return was 10.  

We can also see that the outcome of the first coin toss was 0 so we lost 1. This means that before the second 

coin toss our bet size will be 2. The outcome of the second coin toss was 0 so our return is -2. This means

that before the third coin toss our bet size will be 3. The outcome of the third coin toss was 1 so our return is +3.

The reasoning continues in such a way until the last period.  

We can now plot our return distribution if we play the coin toss game 100 times simulated 1000 times.