KEY TAKEAWAYS
Fibonacci and D’Alembert are staking plans, and the math is the same for both: neither one changes the house edge. Each bet still has the same expected loss per dollar staked, so the expected loss over a session equals the total amount wagered multiplied by the house edge. On an American roulette wheel, betting red at 5.26%, a plan that wagers 2,711 units on average over 100 bets loses about 142.7 units on average, while flat betting of 100 units loses about 5.26 units.
What the plans do change is the shape of the results. A Fibonacci plan finishes ahead in about 89.5% of 100-bet sessions in the calculation below, yet its average result is still negative, because a small share of sessions ends in very large losses. D’Alembert is gentler, but the same logic applies.
This guide explains how each plan sets its stakes, gives the formulas for streak losses and expected loss, works an eight-bet example, and compares both plans with flat betting in a table. The numbers are model results for one stated game, not forecasts of any session.
How the Fibonacci and D’Alembert Systems Work
Both plans are applied to even-money bets such as red or black. The player picks a base unit, say $10, and changes the stake after each result according to a fixed rule.
The Fibonacci plan uses the Fibonacci sequence as its stake ladder: 1, 1, 2, 3, 5, 8, 13, 21 and so on, where each number is the sum of the two before it. After a loss the player moves one step up the ladder. After a win the player moves two steps back, or returns to the start if there are fewer than two steps to go back.
The D’Alembert plan adds one unit after a loss and removes one unit after a win, never going below the base stake. Its stakes climb linearly, 1, 2, 3, 4 and so on, rather than at the faster Fibonacci rate.
Both plans are marketed on the idea that wins and losses must even out, so a loss should be followed by a larger stake to recover it. That reasoning is the gambler’s fallacy: each spin is independent, and the wheel does not owe the player a win. As the Martingale system guide shows for doubling stakes, the plan can only decide how much is wagered on each spin, not what the spin does.
The Formulas Behind the Systems
Three results describe both plans. The first two give the cost of a losing streak, and the third gives the expected result.
where:
n = the number of consecutive losses, counted in base units.
F(n + 2) = the (n + 2)th Fibonacci number, with F(1) = 1, F(2) = 1 and F(k) = F(k − 1) + F(k − 2).
Expected total wagered = the average of the sum of all stakes over the session, which depends on the plan.
House edge = the expected loss per unit staked: 2 ÷ 38 ≈ 5.26% for red on an American wheel.
The third formula follows from one fact: each stake is fixed before its spin is played, using only earlier results. The spin itself is independent of that choice, so its expected result is −stake × 5.26%. Adding those expectations across all bets gives the formula, whatever rule picked the stakes.
The assumptions are an American wheel, even-money bets with fixed win probability 18 ÷ 38, a fixed number of bets, no table limit and an unlimited bankroll. Real tables have maximum bets and real bankrolls are finite, which makes the large-loss tail worse than the model suggests.
Worked Example: Eight Bets and a Losing Streak
Suppose a base unit of $10 and eight bets on red at an American wheel, with the results lose, lose, win, lose, win, win, lose, win. That is four wins and four losses.
Step 1. Flat betting: four wins of +1 and four losses of −1 gives a net result of 0 units.
Step 2. Fibonacci stakes: 1, 1, 2, 1, 1, 1, 1, 1. Results: −1, −1, +2, −1, +1, +1, −1, +1, a net of +1 unit, or +$10.
Step 3. D’Alembert stakes: 1, 2, 3, 2, 3, 2, 1, 2. Results: −1, −2, +3, −2, +3, +2, −1, +2, a net of +4 units, or +$40.
Both plans finished ahead on this run, but only because wins and losses were equal in number. The luck of one sequence says nothing about the average, as discussed in variance in gambling.
Now the bad case. Suppose eight straight losses. The chance is (20 ÷ 38)⁸ ≈ 0.589%. Flat betting loses 8 units. Fibonacci loses F(10) − 1 = 55 − 1 = 54 units, or $540. D’Alembert loses 8 × 9 ÷ 2 = 36 units, or $360. A longer streak costs far more than a small win earns back, and the next Fibonacci stake would be 34 units.
Comparing the Systems Over 100 Bets
The table below comes from an exact calculation over every possible sequence of 100 bets on red at an American wheel, starting each plan at one unit, with no table limit and no bankroll limit.
| Plan | Expected total wagered (units) | Expected net result (units) | Chance of finishing ahead | Chance of losing 1,000+ units |
|---|---|---|---|---|
| Flat, 1 unit | 100.0 | −5.26 | 26.50% | 0% |
| D’Alembert | 726.4 | −38.23 | 49.69% | ≈ 0.004% |
| Fibonacci | 2,711.4 | −142.70 | 89.53% | ≈ 0.22% |
Check the third formula against each row. Flat: 100 × 5.26% = 5.26. D’Alembert: 726.4 × 5.26% ≈ 38.23. Fibonacci: 2,711.4 × 5.26% ≈ 142.70. The ratio of expected loss to expected wagered is 5.26% for every plan.
The plans differ in how often they finish ahead. Fibonacci finishes ahead most often and still has the largest average loss, because many small wins are paid for by rare, very large losses. In dollars at a $10 unit, the Fibonacci average is about −$1,427 per 100 bets, against about −$53 for flat betting.
The same comparison applies to any stake rule. Proportional staking such as the Kelly criterion is a different case, because it assumes a positive edge, which casino bets do not provide.
Common Mistakes About Stake Plans
The first mistake is judging a plan by how often it wins. A plan that finishes ahead in 89.5% of sessions can still have a negative average, because the frequency of winning sessions and the size of losing sessions are different quantities.
The second is believing a streak makes a win due. After eight reds in a row, the chance that the next spin is red is still 18 ÷ 38 ≈ 47.37%, exactly as before.
The third is ignoring table limits. A plan that needs a 34-unit bet after eight losses may hit a maximum stake, and then the recovery logic breaks, so the table above is a guide to the average, not to every real table.
The fourth is confusing a lower edge with a gentler plan. D’Alembert grows stakes more slowly, so its swings are smaller, but its edge is the same 5.26%. A smaller swing is a smaller average amount wagered, not a smaller percentage cost. The risk of ruin guide shows how the bankroll limit turns these swings into ruin.
Quick Reference and Check Your Answer
| Item | Fibonacci | D’Alembert |
|---|---|---|
| After a loss | Move one step up | Add 1 unit |
| After a win | Move two steps back | Remove 1 unit |
| Lost after 8 straight losses | 54 units | 36 units |
| Lost after 10 straight losses | 143 units | 55 units |
| Edge on American red | 5.26% | 5.26% |
To check any figure, rebuild the stake list and add it. For Fibonacci, 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 54, matching F(10) − 1. For D’Alembert, 1 + 2 + … + 8 = 36, matching 8 × 9 ÷ 2. Then confirm that expected loss divided by expected wagered equals the house edge. If the two do not match, the stake list or the rounding is the first place to look. State the unit, the wheel type and the number of bets next to any result, since all three change the figures.
Related Concepts and Next Steps
The doubling plan is analyzed in the Martingale system guide, and the edge used in every example here is derived in the roulette house edge article. For what happens when a bankroll is finite, read risk of ruin.
The central point carries over to every staking plan: the stake rule can reshape results but cannot change the average cost of the game. Gambling is best treated as paid entertainment, with the house edge as its price. A reader who wants to continue can use the same method on another game: find the edge, estimate the total wagered, and multiply the two to get the average cost of the session.
Frequently Asked Questions
