Betting Systems Tested: Fibonacci, D'Alembert, Labouchère and Paroli

Every betting system tells you when to bet more and when to bet less. None of them changes the house edge on a single bet, so none of them changes the expected loss: the edge times everything you wager. What systems do change is the shape of your results, trading frequent small wins for rare big losses, or the other way around. Here is the math on four famous ones.

The one equation every system must obey

On a European wheel, every bet has an expected value of −1/37 of its stake, about −2.70%. That is true for the first bet of the night and for a bet placed after ten losses. The wheel does not know your system.

Expected values add, so the expected result of any sequence of bets is the sum of their individual expected values.

This is not an opinion about systems; it follows directly from how expected value works, and mathematicians have proved the general version: no rule for sizing or timing bets can turn a sequence of negative-expectation bets into a positive one. A system that raises your average bet raises your expected loss. One that lowers it lowers your expected loss. That is the full list of options.

All examples below use European even-money bets (red/black, odd/even, high/low) with a $10 unit. Each wins with probability 18/37 ≈ 48.65% and loses with probability 19/37 ≈ 51.35%.

Fibonacci: a slower Martingale

The rules: bet along the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, where each number is the sum of the two before it. After a loss, move one step forward. After a win, move two steps back.

The bets grow more gently than the Martingale's doubling. Six straight losses cost $10 + $10 + $20 + $30 + $50 + $80 = $200, and the next bet is $130. Ten straight losses cost $1,430, because the first ten Fibonacci numbers add to 143.

The catch is the recovery. A win only covers roughly the last two bets. After those six losses (−$200), a $130 win brings you to −$70. Step back to $50 and win: −$20. Step back to $20 and win: $0. You need three straight wins just to break even, and a fourth to make your $10.

Fibonacci is gentler than the Martingale on the way down and slower on the way back. Its expected loss is still 2.70% of everything it wagers.

D'Alembert: the equilibrium illusion

The rules: start at one unit. Add one unit after a loss; subtract one unit after a win, never going below one.

The pitch is clever. Because the bet rises by one unit after each loss and falls by one after each win, a session with equal numbers of wins and losses always ends ahead by one unit per win. Ten wins and ten losses, in any order, net +10 units, as long as the bet never hits the one-unit floor. The system takes its name from the 18th-century French mathematician Jean le Rond d'Alembert, and it rests on the hope that wins and losses even out.

Roulette does not balance. With 18 wins for every 19 losses on average, losses pull ahead by about one bet in every 37. Over 370 spins you expect 190 losses and 180 wins. On top of that drift, random swings grow with the square root of the number of spins. When losses get ahead, your bets climb, so you wager the most exactly when you are behind.

Six straight losses cost $10 + $20 + $30 + $40 + $50 + $60 = $210, and the next bet is $70. That is milder than Fibonacci's next bet, but a long uneven stretch can push bets steadily upward for hours.

Labouchère: the cancellation line

The rules: write a line of numbers, say 1-2-3-4, which adds to your target of 10 units. Bet the first number plus the last. If you win, cross both off. If you lose, write the amount you lost at the end. When every number is crossed off, you have won 10 units.

The appeal is that each win removes two numbers while each loss adds only one. On a long line you can finish while winning only a little more than one bet in three.

The cost shows up in the bet sizes. Starting from 1-2-3-4 with a $10 unit, six straight losses look like this: bets of 5, 6, 7, 8, 9 and 10 units, a loss of 45 units or $450. Your line is now 1 through 10, it adds to 55 units, and your next bet is 11 units, $110. Each further loss makes the line longer and the bets bigger.

Paroli: the reverse approach

The rules: double your bet after each win, up to three wins in a row, then reset. After any loss, reset to one unit.

Paroli flips the shape of the negative progressions. Your losses are always one base bet, and you only press winnings. A cycle ends in one of four ways:

• Lose the first bet: −$10.
• Win, then lose: +$10 − $20 = −$10.
• Win, win, then lose: +$10 + $20 − $40 = −$10.
• Win three times: +$10 + $20 + $40 = +$70.

Paroli is the gentlest system on your bankroll, since a losing streak costs only base bets. The winnings you press are not house money, though. Once you win them, they are your money, and they carry the same 2.70% edge as the rest.

Side by side: what six straight losses cost

Six losses in a row has probability (19/37)^6 ≈ 1.83% from any particular spin. That sounds rare, but across a session it is not. Working through every spin exactly, a run of six or more straight losses appears in about 60% of 100-spin sequences and about 84% of 200-spin sequences.

The last column is the point. The systems differ wildly in how much damage a bad streak does, and not at all in what they cost per dollar wagered.

What systems actually change

• Negative progressions (Martingale, Fibonacci, D'Alembert, Labouchère) raise bets after losses. They produce a high chance of a small win and a small chance of a large loss, and they tend to increase total wagering when you are behind.
• Positive progressions (Paroli) raise bets after wins. They produce a high chance of a small loss and a small chance of a moderate win.
• All of them rely, at least in their marketing, on the idea that a result is due. It never is; see the gambler's fallacy.

The bottom line

Betting systems are ways of rearranging risk, not reducing it. Fibonacci, D'Alembert and Labouchère smooth out the Martingale's cliff but keep its basic shape: many small wins, occasional big losses. Paroli inverts it. Every one of them loses 2.70% of the money it puts through a European wheel, and 5.26% on an American one.

If you enjoy a system's rhythm, the honest way to use it is as entertainment with a fixed budget. If you are using one to recover losses, stop and read the responsible gambling guide.