Betting systems in roulette: Martingale, Fibonacci, D'Alembert — why they don't work in the long term (2026)

Roulette betting systems are an interesting chapter in gambling history. They've been around for the last 300 years, bearing the names of French mathematicians and English aristocrats, all promising the same thing: a way to beat the casino without knowing the future. Some have spread so massively among players over the 300 years that even a complete beginner knows them today — Martingale as "double down after a loss", Fibonacci as "advance in a series of 1, 1, 2, 3, 5, 8, 13".

This article is not intended to prohibit you from using these systems. They are fun, structured and can make a roulette night more predictable. The goal is to mathematically demonstrate why none of them can beat the house edge, why the table limit guarantees bankruptcy even before the "system comes back", and why you shouldn't put more than a reasonable entertainment budget into any of them. For a deeper context of the offer and rules in New Zealand online roulette, I recommend first going through the articles roulette rules and strategies and roulette winning tips .

The basics: why no system beats roulette

European roulette has 37 numbers: 18 red, 18 black and one green zero. When betting on red or black, the probability of winning is 18/37= 48.65%. The payout is 1:1, so the expected value (EV) of one bet is:

EV= (18/37 × 1) − (19/37 × 1)= −1/37 ≈ −2.70%

This means: for every NZ$ 0bet, the player loses roughly 67 pennies on long-term average. For American roulette (with two zeros, 38 fields) the calculation is the same:

EV= (18/38 × 1) − (20/38 × 1)= −2/38 ≈ −5.26%

The betting system is just a rule, according to which you determine the amount of the bet for the next spin based on the previous result. But the expected value of the sum of independent bets= the sum of their expected values. Whether you bet 100 times at NZ$ 0or NZ$ 0or successively NZ$ 4010, the total EV will always be −2.70% of the total amount bet. No amount change rule changes this fundamental thing.

What the system changes: distribution of results (variance). Some systems give many small wins and occasionally a huge loss. Other steady slow decline of bankroll. Some give big wins rarely. These are all legitimate reasons to choose a system — some players are comfortable with a certain type of distribution. What will not change, however, is the long-term expected outcome. It remains negative.

A detailed mathematical analysis of the casino's dominance in table games (including all types of roulette bets) is discussed in the article house edge in live casino games .

Martingale: the most famous and most dangerous system

Rule

Start with a base bet (e.g. NZ$ 0) on an even-probability bet (red/black, even/odd, 1 — 18/19 — 36). After a win, start again with the base bet. After losing, double the bet.

Origin: 18th century, France. Philosophy: one win covers all previous losses and adds a profit equal to the base bet.

Example sequence

• Spin 1: NZ$ 0on red → black fell. Loss -NZ$ 0
• Spin 2: NZ$ 0on red → black fell. Loss -NZ$ 0
• Spin 3: NZ$ 0on red → black fell. Loss -NZ$ 10
• Spin 4: NZ$ 10on red → red fell. Profit +NZ$ 10Cumulative result: +NZ$ 0
• Spin 5: NZ$ 0— the sequence starts from the base bet.

The lure of Martingale: for 4 spins the player won NZ$ 0although 3 out of 4 spins were in the red. It looks invincible — until the player goes on a long losing streak. In european roulette, the probability of a different color (or zero) falling 7 times in a row is 0.94% — that is, once in about 106 sessions. For 8 in a row 0.48%. For 10 in a row 0.13%. These numbers sound small, but statistically they will definitely come if you play long enough.

With a base bet of NZ$ 0and 10 losses in a row, the player must bet NZ$ 0for the next spin — just to restore the original 25 crown profit. With 15 losses already NZ$ 40per bet. These amounts hit the wall for two reasons: the casino's table limit (typically 25,000 — NZ$ 0per bet) and the financial reality of the player's account.

Important math: Martingale does not guarantee a win, only bankruptcy. The question is not "if" we get to 8. Or 10. Loses in a row, but "when". With 200 sessions of 50 spins with Martingale, such a streak is almost certain. And when it comes, it destroys everything the player has previously acquired — plus a significant portion of the bankroll.

Fibonacci: slower progression, same result

Rule

Bet according to the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... After losing, go to the next member of the series. Go back two members after winning.

Origin: 13th century, Leonardo Pisano (Fibonacci). Gambling applications only in the 19th and 20th centuries. The philosophy: a gradual increase in the bet that survives more losses than a Martingale.

Example sequence

• Spin 1: NZ$ 0→ lose (row: 1, 1). Loss -NZ$ 0
• Spin 2: NZ$ 0→ lose (row: 1, 1, 2). Loss -NZ$ 0
• Spin 3: NZ$ 0→ lose (row: 1, 1, 2, 3). Loss -NZ$ 0
• Spin 4: NZ$ 0→ lose (row: 1, 1, 2, 3, 5). Loss -NZ$ 10
• Spin 5: NZ$ 0→ win +NZ$ 0(row: 1, 1, 2). Cumulative: -NZ$ 0
• Spin 6: NZ$ 0→ win +NZ$ 0(row: 1). Cumulative: €.
• Spin 7: NZ$ 0— back to base bet.

Fibonacci sounds more elegant and has a more solid mathematical foundation, but mathematically it is just as problematic as Martingale. After 12 losses in a row, the player must bet € (vs. Martingale NZ$ 0). Slower growth allows you to survive longer — but also means that after winning, the player will not get a full comeback, but only a step back two ranks.

So Fibonacci is the Martingale "adjusted to a more considerate curve" — but it still leads to the same problem. The main difference: with Fibonacci, the player does not get the equivalent of the entire losing streak when he wins, only a partial comeback. This means slower bankroll recovery, but also lower absolute profits during a successful streak.

D'Alembert: the mathematical myth of equilibrium

Rule

After a loss, increase the bet by one unit (e.g. +NZ$ 0). After a win, reduce the bet by one unit.

Origin: 18th century, French mathematician Jean le Rond D'Alembert. Philosophy: long-term "equilibrium" outcomes — the idea that after losing, winning is more likely.

Example sequence (base bet NZ$ 0)

• Spin 1: NZ$ 0→ lose. Status: -NZ$ 0Bet NZ$ 0
• Spin 2: NZ$ 0→ lose. Status: -NZ$ 10Bet NZ$ 10
• Spin 3: NZ$ 10→ win. Balance: -NZ$ 0Bet NZ$ 0
• Spin 4: NZ$ 0→ win. Status: +NZ$ 0Bet NZ$ 0
• Spin 5: NZ$ 0→ lose. Status: -NZ$ 0Bet NZ$ 0

D'Alembert is much healthier from a bankroll perspective than Martingale or Fibonacci. The bankroll lasts a long time, the stakes grow linearly. The problem: the whole system is based on a faulty premise. D'Alembert suggested that after a losing streak, a win was "more likely" because events "must even out". This is the gambler's fallacy, one of the most famous cognitive biases in gambling psychology.

The roulette wheel is memoryless. The probability of landing red is 18/37 on each spin regardless of what came before. After 10 losses in a row the probability is 11. Wins still 18/37= 48.65%, no more. The analogy from the article breaks it down in detail the volatility of semi-healing machines — we're talking about RNG there, but the principle is the same for both physical and online roulette wheels.

So from a practical point of view, D'Alembert reduces the variability of results and provides a slow, predictable decline in bankroll. For some, this is a more tolerable experience than the Martingale rollercoaster, but the result after 1,000 spins remains the same: -2.70% of the total bet.

Labouchere (cancellation system): the illusion of control

Rule

Write a series of numbers on paper — for example 1, 2, 3, 4. The bet is the sum of the first and last number (1 + 4= 5). After winning, cross out these two numbers. After a loss, add the amount of the bet to the end of the row (5). Goal: get to an empty row when you return with the original win.

Origin: 19th century, British politician Henry Labouchere. Philosophy: a structured way of "wiping out" numbers with a planned profit equal to the sum of the original series.

Labouchere has one fascinating feature: for 50% chance bets, all it takes is for the player to win at least half of the spins, and the whole line will explode. This may seem like a fundamental breakthrough — in fact, it's just a different distribution of EV, not a change. In the long run, the player will find himself in a situation where a losing streak will extend the streak to 15, 20 or more numbers and the stakes will grow to an unmanageable size.

Example: if you start with a streak of 1-2-3-4 and successive losses come, the streak grows: 1-2-3-4-5, 1-2-3-4-5-6, 1-2-3-4-5-6-7. After 7 losses in a row, the bet would be 1+7= 8. However, if the streak continues, the bets double — especially when you start to "smear" from the outside and the high numbers remain in the middle.

In practice, Labouchere combines Martingale with manual registration — for some players, it's a way to "have something to do" at roulette outside of spinning the wheel. However, from a mathematical point of view, it does not beat either the neutral game or roulette with a house edge of 2.70%.

Paroli (reverse Martingale): safe brother

Rule

Place the base bet. Double after winning. After the second win, double once more. After the third win (or the agreed maximum), return to the base bet. Immediately return to base bet after losing.

Origin: 16th — 17th century, Italy. Philosophy: "go for the wins, don't deepen the losses". Maximizes short-term winning streaks.

Paroli is the mildest of the described systems in terms of bankroll. The bankroll cannot be destroyed quickly because after each loss you return to the base bet. However, in the long run, Paroli is still subject to the same house edge as any other system. The difference is only in the distribution of results: Paroli gives several medium-sized wins (after 3 wins in a row when doubling, the player earns 7 times the base bet) alternated with a lot of small losses.

For a player who wants to spend time playing roulette with a low risk of a major draw, Paroli is a reasonable choice. Not as a way to "win" though — but as a way to spread the bankroll evenly throughout the evening.

New Zealand casino table limits and when the system crashes

For Martingale or Fibonacci to have at least a theoretical chance of working, players would have to have an infinite bankroll and an infinite table limit. The reality is different. For online and live roulette, you will encounter different limits depending on the type of table, VIP status and game provider:

The ranges are model and serve to understand the risk of progression. Always check the actual limit directly at the table before starting the game.

With a basic bet of NZ$ 0and a standard max limit of NZ$ 0the player hits the ceiling already at the 10th Martingale doubling (NZ$ 0exceeds the limit). In other words: in European Roulette, with a probability of 0.13%, 10 losses will occur in a row and the player must pay the full loss of NZ$ 0without the possibility of calling. With 50 sessions per year of 100 spins, this situation happens on average once or twice.

With a base bet of NZ$ 0(where Martingale "earns" faster) you will hit the limit after 8 losses in a row — a probability of 0.48%. With 100 sessions a year, this happens 4-5 times a year, and each such event means a loss of NZ$ 0The arithmetic remains relentless.

Simulation of 1000 Martingale sessions on European Roulette

To make the math concrete, we simulate 1,000 independent Martingale sessions with the following parameters:

• European roulette, one zero (37 numbers).
• Bet on red/black only (probability of winning 18/37= 48.65%).
• Basic bet NZ$ 0
• Maximum table bet NZ$ 0— if exceeded, the session ends as a "Martingale Bankruptcy".
• Player's bankroll NZ$ 0— if exceeded, the session ends as "bankroll loss".
• Goal of the session: win € (10 spins in addition to the base bet).

Values ​​are simulation based on a Monte Carlo model for 1,000 independent sessions. Specific results may vary slightly with a different start of the pseudorandom simulation.

Summary: 1000 sessions × (87.1% × € − 7.8% × NZ$ 0− 5.1% × NZ$ 0)= € − NZ$ 60− NZ$ 40= −NZ$ 30So the average loss per session is NZ$ 1000The 87% of "successful" sessions do not beat the 13% that fail. This is the exact mathematical reality of the system.

Confirmation bias: After 50 successful sessions, the player will feel that the system is working. Social networks and YouTube videos show just these "successful" moments — few people make a video of a session where they lost NZ$ 0Statistically, however, those 13% of disastrous sessions account for 78% of all money lost. This is why the Martingale (and all its derivatives) returns exactly the expected loss of 2.70% of the bet in the long run.

What can you really take away from the systems?

Roulette betting systems are not "bad" in the sense that they break the rules of the game or get you into trouble in New Zealand licensed casinos. They are legal, popular, and often part of gaming lore. The real question is: how to approach them without illusion.

• A system as an organization of play, not as a winning strategy. If it helps you have structure — how much to bet, when to leave — it has value.
• Always with a fixed stop-loss limit. Before you start the Martingale, determine the amount at which you will stop (typically 50% of the budget). Without this limit, a 100-session win can be wiped out by one bad day.
• Only play with money you are prepared to lose. No system guarantees profit. Any amount you "need back" does not belong to roulette.
• European roulette always before American roulette. The difference of 2.70% vs. 5.26% is almost double. For systems, this difference accumulates quickly.
• For a longer session, try regressive systems. D'Alembert or Paroli are less stressful and will keep the bankroll longer than Martingale.
• Self-limiting limits are more effective than any system. Hard limits through the casino app will prevent a catastrophic loss better than any "let's say we quit after losing€" rule. Details in the article self-limiting limits in New Zealand online casinos .

The most honest sentence about betting systems is: "I want to play roulette with a structure and I know that in the long run I will lose 2.70% of the bet". If you approach it this way, no disaster will surprise you.

Useful resources:

• Online Roulette — Rules, Variants and Operators
• Tips for winning at roulette — practical advice
• House edge in live casino games
• New Zealand's best online live casinos
• Semi-heel volatility (same math as gambler's fallacy)
• Responsible gaming — how to prevent the development of problem gambling
• Self limiting limits in New Zealand online casinos

❓ Frequently asked questions about betting systems in roulette

Online Roulette New Zealand House edge in live games