The Fibonacci system is a well known system in roulette. Its a progressive
system that uses Money Management to try to beat losing streaks:
It is generally applied to the 'even' bets such as Red/Black, Even/Odd, Low/High. Where you have 18/37 = 48.6% chance on an individual win.
The philosophy is similar as the Martingale. With the Martingale the philosophy behind the system is that you make a bet equal to ALL your losses, and
hope to win on the next spin.
With the Fibonacci the philosophy behind the system is that you make a bet equal to THE LAST 2 losses, and hope to win on the next spin.

The principle is to bet $1 on Red or Black, and then when you lose you bet $1 again. If you lose again you bet the losses of your last 2 bets: $1 +$1 = $2,
if lose again, you bet the losses of your last 2 bets: $2 +$1 = $3. By this time you lost total $7. The next bet is $3 + $2 = $5 .
This gives a progression called Fibonacci series: $1, $1, $2, $3, $5, $8, $13, $21, $34, $55, $ 89, .....

Depending on what you do what you win, you can have 3 different ways of playing:

1) When you win, you go back only 1 step in the progression, so in the example above you bet $3. This way IF you win 2 consequtive spins
you won ALL your losses back and have $1 gain.

2) When you win, you go back 2 steps in the progression, as you won back the last 2 losses. So in the example above to $2, and you continue try to win back the 2 losses
of an older bet.

3) When you win, you go back only 1 step in the progression, so in the example above you bet $3. This way IF you win on the next spin
you won ALL your losses back. And then you start the progression from $1.

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The graph below shows the bankroll after 18 spins using the Fibonacci progression described above under 1) and betting on Red/Black.
As always, starting with $100 bankroll and $1 as first bet.

If you lose 8x, you go bankrupt. And there is about 3% chance that this happens. While due to the progression there is also a very small chance that you end up at $161.
Most likely though, you end between $90 and $120.

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