Roulette strategies tested: Martingale, Fibonacci and d'Alembert

No betting system changes the house edge in roulette. On a European wheel you lose on average EUR 2.70 of every EUR 100 you bet, whatever pattern you bet in. What a system does change is the shape of your results: many small wins and a rare large loss, or the other way round. We ran the numbers on the three best-known systems - Martingale, Fibonacci and d'Alembert - so you can see both halves of that sentence: why they feel like they work, and why they do not.

Why no system beats the zero

The zero pays nobody on an even-money bet, and that one pocket is the casino's edge on every spin.

A European wheel has 37 pockets: 18 red, 18 black and the zero. A bet on red wins on 18 pockets and loses on 19. Over time you get back EUR 36 for every EUR 37 wagered - a house edge of 1/37, or 2.70%. On an American wheel with an extra double zero, the edge is 2/38, or 5.26%. French roulette with the La Partage rule returns half of an even-money stake when the zero lands, which halves the edge on those bets to 1.35%. These figures are standard; Wizard of Odds lists the same numbers.

The key point is that every bet on the table carries the same edge. If each single bet loses 2.70% on average, any sequence of bets - doubled, increased, decreased - also loses 2.70% of the total amount wagered. A system decides how much you bet and when. It cannot decide what each bet is worth. How the house edge turns into a game's return to player is explained in our article on RTP and volatility, and the rules and bets of the game itself are on our roulette page.

Martingale: the test

Martingale doubles the stake after every loss, so the first win recovers everything lost plus one base unit - until a losing streak or the table limit stops the doubling.

It is the most popular system because it looks unbeatable. You bet EUR 1 on red. If you lose, you bet EUR 2, then EUR 4, then EUR 8. When red finally comes, the win covers all the previous losses and leaves you EUR 1 ahead. Then you start again at EUR 1.

The catch is how fast the stakes grow. On a European wheel the chance of losing a single even-money bet is 19/37, and the chance of losing several in a row is that number multiplied by itself:

• 5 losses in a row: 3.57% (1 in 28). Your next stake is EUR 16, and you have lost EUR 31.
• 7 losses in a row: 0.94% (1 in 106). Next stake EUR 64, lost EUR 127.
• 9 losses in a row: 0.25% (1 in 403). Next stake EUR 256, lost EUR 511.
• 10 losses in a row: 0.13% (1 in 784). Next stake EUR 512, lost EUR 1,023.

Now add the table limit. Every table has a maximum bet, set by the casino. At a table with a EUR 500 maximum, starting from EUR 1 you can double nine times: the tenth stake, EUR 512, is not allowed. Nine losses in a row end the series at minus EUR 511. That one loss wipes out 511 winning series.

A streak of nine sounds rare, and in a single series it is. But you play many series in a session. We calculated it exactly and checked it with a simulation of 200,000 sessions: with a EUR 1 start and a EUR 500 limit, the wall is hit in 10.7% of 100-spin sessions and in 45.3% of 500-spin sessions.

And here is the result that explains why people believe in Martingale. We simulated 400,000 sessions of 100 spins each. Betting a flat EUR 1 on red, 35.5% of sessions ended ahead. With Martingale, 87.8% of sessions ended ahead. Yet across all sessions Martingale still lost 2.7% of everything wagered - exactly the house edge. Most sessions win a little; a few lose a lot; the average does not move.

Table limits and your bankroll

The bigger the table limit, the rarer the wall - and the bigger the loss when it comes.

Table limits differ from casino to casino and from table to table, from a few hundred to tens of thousands of euros. EUR 500 below is an example of a low-limit table, not the limit of any particular casino.

• Start EUR 1, limit EUR 500: 9 doublings possible. A failed series costs EUR 511. Chance of hitting it within 100 spins: 10.7%.
• Start EUR 1, limit EUR 2,000: 11 doublings, a failed series costs EUR 2,047, chance within 100 spins 2.9%.
• Start EUR 1, limit EUR 5,000: 13 doublings, a failed series costs EUR 8,191, chance within 100 spins 0.7%.
• Start EUR 5, limit EUR 500: 7 doublings, a failed series costs EUR 635, chance within 100 spins 36.3%.

In practice most players run out of money before they reach the table limit: surviving nine losses in a row takes EUR 511 on hand, to win EUR 1.

Fibonacci

Fibonacci moves one step up a number sequence after a loss and two steps back after a win, so stakes grow more slowly than with Martingale - and a win no longer recovers everything.

The sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, where each number is the sum of the two before it. A short run in euros: bet 1, lose (-1); bet 1, lose (-2); bet 2, lose (-4); bet 3, lose (-7); bet 5, win (-2); step back two places and bet 2, win (0). Two wins after four losses bring you back to zero, not ahead.

The slower growth is the selling point, but long streaks still hurt: after ten losses in a row the next stake is EUR 89, and you are EUR 143 down. In our simulation, Fibonacci lost 2.68% of the total amount wagered - the house edge again, within the margin of the simulation.

d'Alembert

d'Alembert adds one unit to the stake after a loss and takes one away after a win, on the belief that wins and losses even out.

A short run: bet 1, lose (-1); bet 2, lose (-3); bet 3, win (0); bet 2, win (+2). Two wins and two losses left you two units ahead, and that is exactly why the system feels like it works. But it only works if wins and losses balance, and on a European wheel they do not: an even-money bet has 18 winning pockets against 19 losing ones. In our simulation d'Alembert ended ahead in 58.9% of 100-spin sessions and lost 2.74% of the total wagered.

Labouchère and Paroli in brief

Both change the pattern of stakes, and neither changes the edge.

Labouchère starts with a line of numbers, for example 1-2-3. You bet the first and last number added together; a win crosses both off, a loss adds the lost stake to the end of the line. A losing run makes the line and the stakes grow quickly.

Paroli, also called reverse Martingale, doubles after a win instead of a loss, usually for three wins in a row, then goes back to the start. It risks only the base stake in each series and trades many small losses for occasional larger wins. The expected loss is still 2.70% of everything wagered.

What actually changes the outcome

The table you choose, not the pattern you bet.

The only lever a roulette player really has is the rule set. French roulette with La Partage has an edge of 1.35% on even-money bets, European roulette 2.70%, American roulette 5.26%. On the same EUR 1,000 wagered on red, that is an expected loss of EUR 13.50, EUR 27 or EUR 52.60.

Whichever casino you choose - for example Spinit, FS Casino or Spinanga - open the roulette lobby and look for "French" or "La Partage" in the table name, and avoid double-zero tables. Live dealer tables are covered in our piece on live casino shows.

The other lever is the one no system gives you: a limit. Decide your session budget before you sit down and stop when it is gone, win or lose. A doubling system is the fastest way roulette offers to one large loss. Our article on responsible gambling tools shows how to set deposit and loss limits, and self-exclusion in Latvia explains how to block yourself from gambling sites altogether. More help is on our responsible gambling page. Gambling is for adults aged 18 and over.