What is Fibonacci, and what does it have to do with roulette?
The Fibonacci sequence is a list of numbers which, in the form we will use here, begins like this:
1, 1, 2, 3, 5, 8, 13, 21…
From the third number onwards, each is obtained by adding the two before it. One plus one gives two. One plus two gives three. Two plus three gives five. And the list continues in the same way.
That rule tells us the next number in the sequence. It does not tell us the next result on a roulette wheel.
Roulette has a wheel with numbered pockets and a ball that lands in one of them. Each release of the ball that produces a result is a spin. A European wheel has 37 pockets: 18 red, 18 black and one green zero.
A bet on red depends on the ball landing in a red pocket. Under the rules used in our examples, a win returns the amount placed and adds a profit equal to that amount. If the result is black or zero, the entire amount placed is lost.
The Fibonacci system uses the numbers in the sequence to change those amounts. It applies the list to bets; it is not a property of the wheel or a way to predict results.
How the model in this article works
We will use units to keep track of the figures. A unit is an abstract measure, not a recommended amount of money.
The version examined here follows these rules:
- It starts at the first position in the list, with one unit.
- After a loss, it moves to the next position.
- After a win, it moves back two positions.
- If that move would go beyond the start, it stays at the first position.
The two initial ones occupy different positions. After the first loss, the model moves from the first 1 to the second 1. The amount is still one unit. Only after another loss does it reach 2.
It is also important to distinguish positions from amounts. Moving back two positions from 8 leads to 3: first past 5 and then to 3. It does not mean subtracting two units from eight.
All the examples assume a balanced European wheel, an equal chance of landing in each pocket and independent spins. Red remains the favourable result; black or zero means a full loss.
Other versions of the system and other rules for zero exist. We fix these conditions here so the figures can be followed. Each example ends after the stated number of spins; it does not assume that play can continue until losses are recovered.
What happens after several losses and one win
Imagine four consecutive losses.
The first loses one unit. The second loses another. The third loses two and the fourth loses three. Seven units have been lost in total.
The next amount in the sequence is five. This time the result is favourable and produces a profit of five units.
| Spin | Amount wagered | Outcome | Profit or loss | Running balance |
|---|---|---|---|---|
| 1 | 1 | Loss | -1 | -1 |
| 2 | 1 | Loss | -1 | -2 |
| 3 | 2 | Loss | -2 | -4 |
| 4 | 3 | Loss | -3 | -7 |
| 5 | 5 | Win | 5 | -2 |
The win reduces the losses, but the balance is still two units short of the starting point.
The payout on that fifth spin can cause confusion: ten units are received, but five are the return of the amount placed. Only the other five are profit.
After winning with five, the model moves back two positions. From 5 it passes 3 and reaches 2. That would be the next amount.
The rule has reduced what would be at stake, but the balance remains negative. Moving towards the start of the list does not automatically recover the money.
Why moving back two positions does not guarantee recovery
Let us extend the previous example by one more spin, also favourable.
The balance was minus two and the amount was two units. That new win adds a profit of two, bringing the balance to zero.
In this particular case, two wins after four losses return the balance to its starting point. There is no profit: the earlier losses have been recovered.
But that outcome cannot be turned into a general rule.
Now suppose there are six consecutive losses, at amounts of one, one, two, three, five and eight units. The losses total twenty.
The next amount is thirteen. A win at thirteen reduces the loss from twenty to seven. The model then moves back two positions, from 13 to 5.
If it also wins at five, the balance improves but still ends at minus two.
| Fictional example | Earlier losses | Profits from the two wins | Final balance |
|---|---|---|---|
| Four losses and two wins | 7 | 5 and 2 | 0 |
| Six losses and two wins | 20 | 13 and 5 | −2 |
Both examples include two wins at the end. Only the first recovers everything lost.
The figures for a favourable sequence can be correct without guaranteeing recovery in other sequences. Those wins are also assumed outcomes used to explain the figures, not results that must arrive.
How the amounts and accumulated losses grow
The first Fibonacci numbers are small. The repeated one at the start can make the growth seem particularly slow.
However, each loss adds to the earlier ones, and later amounts increase.
| Consecutive loss | Amount lost on that spin | Total lost since the start |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 1 | 2 |
| 3 | 2 | 4 |
| 4 | 3 | 7 |
| 5 | 5 | 12 |
| 6 | 8 | 20 |
After the sixth loss, the last amount was eight units, but the total lost has already reached twenty.
The next step would require another thirteen units to be available. The earlier twenty have already been lost; they cannot be used to take that step.
Looking only at the number the system has reached therefore gives an incomplete picture. An amount of eight does not mean that only eight have been lost.
Nor is it enough to observe that Fibonacci grows more slowly than Martingale’s doubling. The amounts still increase and losses accumulate. That difference in pace does not guarantee that the available money will be enough or that a future win will make up for what happened.
What happens when the sequence cannot be followed
A list of numbers can continue indefinitely. Available money and table conditions cannot.
Imagine a fictional starting balance of fifteen units. After five losses of one, one, two, three and five, twelve have been lost. Three remain.
The next number in the sequence is eight. With only three units left, that step cannot be taken.
The system is interrupted even though the balance has not yet reached zero. Its rule specifies an amount that is no longer available.
Now suppose the situation is different: enough money remains, but the table allows a maximum of five units per bet. The next step of eight would also be impossible.
These are two distinct obstacles. One depends on the balance and the other on the permitted limit.
The examples are not intended to find a starting amount that makes the system work. They show why an explanation assuming it is always possible to continue leaves out an important part of the risk.
Being able to take the next step does not guarantee recovery either. The next spin can be another loss.
Why the sequence does not change the probabilities
The Fibonacci sequence has a rule for calculating its numbers. A balanced wheel does not use that rule to decide where the ball lands.
In our examples, each spin still has 18 favourable pockets for red and 19 unfavourable ones: the 18 black pockets and zero. The chance of red is 18 out of 37, approximately 48.65%.
That percentage does not increase after several losses. It does not change because the amount moves from three to five or from eight to thirteen units either.
Zero explains why red and black do not each have a 50% chance. Although both colours have the same number of pockets, there is another pocket that belongs to neither.
With a profit equal to the amount placed on a win and a full loss on black or zero, the casino’s mathematical advantage is approximately 2.70% of the total amount wagered.
This is not a fee charged on every spin or a fixed loss per session. Some sessions end in profit and others in loss. The figure expresses a mathematical average, not what will necessarily happen in an individual case.
The total amount wagered is not necessarily the starting balance either: the same money can be put at stake several times.
Fibonacci changes the amounts that depend on each result. It does not change the pockets, probabilities or payouts.
How it differs from Martingale, D’Alembert and Paroli
These systems have something in common: they set rules for changing amounts according to previous results. What differs is the rule.
In classic Martingale, the amount doubles after a loss. Several consecutive losses produce amounts of one, two, four and eight units.
In D’Alembert, one unit is added after a loss and one is subtracted after a win, subject to the defined minimum. With consecutive losses, the amounts are one, two, three and four.
In Fibonacci, the model moves forward through the list after a loss and back two positions after a win. The initial amounts during losses are one, one, two, three, five and eight units.
In the Paroli example examined in the other article, amounts increase after wins. It considers a sequence of one, two and four units, ending at the first loss or after three wins.
These differences produce different balances for the same series of results. They do not remove the casino’s advantage.
Comparing them helps explain how the amounts and possible losses change, not identify a system that guarantees a profit.
What recording a session can teach us
A large win can stand out so much that earlier losses are easy to forget. In the six-loss example, winning thirteen units looks like a substantial recovery. It is, in part, but seven units are still lost overall.
Recording what happened helps preserve that context. It distinguishes a favourable spin from a profitable session and prevents the last result from replacing the full picture.
It also helps separate an observed streak from a prediction. Several black results describe what has happened; they do not mean red must appear next. The wheel does not compensate for a list of results to return lost money.
RouletteLab allows spins to be recorded and descriptive statistics, such as frequencies and streaks, to be viewed. That information can be used to compare impressions with recorded results. The application does not execute strategies or calculate the profits or losses in these examples.
Examining Fibonacci helps explain an important distinction: knowing the next amount is not knowing what will happen on the next spin. The sequence orders the system’s numbers, not the results on the roulette wheel.