The Fibonacci is a popular Roulette betting system that is based on a naturally occurring mathematical sequence. The sequence itself is cumulative. In other words, the next number is equal to the sum of the two previous ones. So the first 12 numbers in the sequence are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
How this system works is:
You progress through the sequence on losing bets and return towards the start with winning bets. Each time you lose, you move on to the next number in the sequence. Each time you win, you step back two numbers.
I was wondering, if I were to use £10 as my initial bet betting on red and black only on the American Roulette wheel (with the double zeros), what are the chances of winning once in a series of 12 bets? That is, what are my chances of winning at least £10 in the total sum of money that I need to put in for the whole series of 12 bets:
£10, £10, £20, £30, £50, £80, £130, £210, £340, £550, £890, £1440
Note that this is different than the Martingale system in that once you win a bet, you don't start over at the beginning of the sequence (£10 initial bet). Instead, you step back two numbers.
Best Answer
It seems that the "American roulette wheel" is one that has $18$ red numbers, $18$ black numbers, and $2$ green numbers (marked "$0$"): a total of $38$ numbers. So for an individual bet on either red or black, the probability of winning that bet is $p = \dfrac{18}{38}$, and the probability of losing that bet is $q = 1 - p = \dfrac{20}{38}$.
As the different bets are independent, the betting strategy used (Fibonacci or otherwise) does not matter for the probability we want to calculate here:
$$\Pr(\text{winning at least once in a series of $12$ bets})\\ = 1 - \Pr(\text{losing all $12$ bets})\\ = 1 - q^{12}\\ = 1 - \left(\frac{20}{38}\right)^{12} \\ = \frac{2212314919066161}{2213314919066161} \\ \approx 0.999548\dots $$
So there's a $99.95\%$ chance that you win at least one of the $12$ bets. Note again that this does not depend on the betting strategy you use. Only the actual amount you'll win depends on the betting strategy of how much money to place on each bet.