Betting Systems Throughout History: What the Mathematics Reveals

Betting systems have been around for centuries. Long before online casino games appeared on PokerStars, players were already trying to plan their bets around winning and losing streaks. Some systems doubled the bet after a loss, others increased it more slowly, while a few took the opposite approach and raised the amount after a win.

Names such as Martingale, Fibonacci and D’Alembert can make these systems sound more mathematical than they really are. They do use set patterns to decide the next bet, but the pattern cannot change the odds of the casino game.

Why Betting Systems Became So Popular

Casino games naturally create streaks. A roulette wheel might land on Red several times in a row, while a Blackjack player can win three hands before losing the next four. When outcomes bunch together like this, it can be tempting to think the next one should somehow be connected to what has already happened.

Betting systems put rules around those streaks. Instead of choosing a new amount from scratch before every round, the player follows a set pattern. A loss might mean doubling the next bet, raising the amount or moving forwards through a number sequence. A win could mean lowering the bet, moving backwards or starting again.

A set pattern gives the player something to follow when the outcomes start to bunch together. That has helped systems such as Martingale and D’Alembert stay familiar long after the games they were first used with changed. The amount can change from one round to the next without changing how the roulette wheel, cards or dice work.

Roulette became closely linked with many of these systems because its even-money bets are simple to follow. Red and Black, Odd and Even and High and Low all pay 1:1, which is why these bets are often used with betting systems. None of them are true 50/50 bets on a standard roulette wheel, though, because of the green zero.

The House Edge Comes Before the Betting System

A European Roulette wheel has 37 pockets. There are 18 red numbers, 18 black numbers and one green zero. A bet on Red wins when the ball lands in one of the 18 red pockets and loses when it lands on one of the 18 black numbers or zero. That gives Red a winning probability of 18/37, or around 48.65%. The chance of losing is 19/37 or around 51.35%. That small difference is enough to create a house edge of roughly 2.70% on a standard even-money bet.

American Roulette adds another green pocket, 00. There are still only 18 red and 18 black numbers, but there are now 38 pockets in total. This raises the standard house edge on an even-money bet to around 5.26%. Changing the amount being bet doesn’t change either figure. A £1 bet on Red and a £100 bet on Red have the same probability of winning. It makes no difference if the £100 bet only became that large because several earlier bets had lost.

Expected Value Explains the Problem

Expected value looks at what can be expected from a bet over a large number of rounds. On European Roulette, even-money bets have a house edge of around 2.70%. Changing the amount after a win or loss doesn’t change that. Martingale, Fibonacci and D’Alembert may change how much is bet from one round to the next, but the house edge stays the same.

The Martingale System: Doubling After a Loss

The Martingale is probably the most famous betting progression of them all. It is generally linked with 18th-century France and became closely linked with games paying even money. The basic rule is simple: after every loss, double the next bet. Once a bet wins, go back to the starting amount.

The system is often explained using Red or Black in roulette because both pay 1:1. The thinking is that one eventual win will recover all the earlier losses and leave a profit equal to the original bet. When the win comes early enough, that is exactly what the numbers show.

How Martingale Works

If the starting bet is £10 on Red and it loses, the next bet becomes £20. Another loss takes the bet to £40, followed by £80, £160, £320 and so on. If the first two bets lose, the player is £30 behind. A £40 win then brings in £40, covering those earlier losses and leaving the run £10 ahead. The same £10 profit is made if the first bet wins straight away.

Over a short run, that can make Martingale look convincing because the bets may stay fairly small for a while. The numbers change very quickly when the losing run carries on. Starting at £10, eight losses in a row would mean losing £10, £20, £40, £80, £160, £320, £640 and £1,280. That adds up to £2,550 already lost before the next bet is even placed. The system now suggests the player place another bet of £2,560 just to try to finish the whole run £10 ahead. This is why a progression that begins with a fairly small bet can reach several thousand pounds really quickly.

Why Martingale Can Seem to Work 

Martingale often leads to a lot of small runs that finish ahead. If the base amount is £10 and ten of those runs finish ahead, that can leave £100 of profit. A run like that can make the system appear to be doing exactly what was expected. The problem comes when a longer losing run happens. Eight losses from a £10 starting bet have already cost £2,550, which is enough to wipe out a very large number of earlier £10 gains.

Finishing ahead several times doesn’t always mean the system is working well. One longer losing run can still wipe out a lot of those smaller gains.  A short run can also look very different depending on how the spins fall. One player might try Martingale for 20 spins without seeing more than two losses in a row, while another could hit a long losing run almost immediately. Both can happen without anything changing in the roulette maths.

The Fibonacci Betting System

The Fibonacci system gets its name from the number sequence it uses: 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. Each number is found by adding together the previous two. The sequence appears in Leonardo of Pisa’s 1202 book Liber Abaci, where it was used in a problem involving the growth of a rabbit population. Similar number patterns were known before this, but the sequence later became known as the Fibonacci sequence.

The number pattern was later used as the basis for a betting system. After a loss, the player moves one place forwards through the sequence. After a win, the system normally moves back two places. The bets increase more slowly than they do with Martingale.

Fibonacci Grows More Slowly

The slower increase is usually what separates Fibonacci from Martingale. There’s more time before the bet reaches very large amounts, and one loss doesn’t immediately cause the next bet to double. If the system is being used on Red at a European Roulette table, however, each bet still has 18 winning pockets and 19 losing outcomes when zero is included. Moving from 1 to 2 to 3 or 5 units has no connection to what the ball will do next.

The D’Alembert System: One Unit at a Time

The D’Alembert system takes an even slower approach. It is linked with the 18th-century French mathematician Jean le Rond d’Alembert and uses simple progression instead of the Martingale’s doubling or the Fibonacci system.

A player chooses a base unit. After a loss, the next bet goes up by one unit. After a win, it comes down by one unit. With a £5 unit, a losing run would go: £5, £10, £15, £20, £25… The numbers rise steadily instead of doubling. This keeps the jumps smaller during a short losing run, although the amount can still grow if the losses keep coming.

D’Alembert Example

Suppose the first four bets lose at £5, £10, £15 and £20. The player is now £50 behind, and the next bet is £25. If the £25 bet wins, the running loss falls to £25. The next bet drops one unit to £20. If that also wins, the remaining loss is £5. The system needs enough wins at the higher amounts to work its way back. Another loss pushes the bet upwards again, so the progression can continue moving in both directions. D’Alembert doesn’t have the sudden jumps found in Martingale, but the £25 bet after several losses is still the same roulette bet that could have been made at the beginning.

The Paroli System: Increasing After a Win

The Paroli system takes the opposite approach to Martingale, Fibonacci and D’Alembert by increasing the bet after a win. It is an older betting system linked to gambling in 16th-century Italy. A common version begins with one unit and doubles after each win. If the player wins three bets in a row, the run finishes and goes back to the base amount. A loss also takes the bet back to the beginning.

With a £10 starting amount, the first win adds £10. The next bet becomes £20 and, if that wins, the running profit reaches £30. The next bet is £40. A third win would take the total profit to £70 before going back to £10. If that £40 third bet loses instead, the first two wins made £30 while the final loss costs £40. This means the complete run finishes £10 behind.

The Labouchère System

The Labouchère system is more complicated because the next bet comes from a written sequence of numbers. It became linked with 19th-century British politician and journalist Henry Labouchère, who said he found the system in a letter written by the Marquis de Condorcet.

It is also known as the Cancellation System. A player starts by writing down a sequence, such as 1 – 2 – 3 – 4. The first and last numbers are added together to decide the next bet. Using this example, 1 + 4 gives a bet of 5 units. If the bet wins, those two numbers are crossed out, leaving 2 and 3. The next bet would again be 5 units. If a bet loses, the lost amount is added to the end of the sequence instead. The aim is to eventually cross every number off the list. Early wins can shorten the sequence quickly, while losses add more numbers and can make the later bets larger.

How the Numbers Can Grow with Labouchère

If the opening line is 1 – 2 – 3 – 4 and that first 5 unit bet loses, a 5 is added to the end, leaving 1 – 2 – 3 – 4 – 5. The next bet becomes 1 + 5, or 6 units. If that loses as well, 6 is added to the line and the next bet becomes 1 + 6, or 7 units. A longer losing run keeps adding bigger numbers to the end. Wins can remove numbers again, but the order of the outcomes can leave the system working with much larger bets than it started with.

The sequence can eventually run into a table maximum or become too expensive to carry on. Even before that happens, the numbers being written down have no effect on how many red, black or green pockets are on the roulette wheel.

Why Winning Streaks Can Make Systems Look Convincing

Short sessions can make almost any betting system look better or worse than it really is. A Martingale player might go through several runs without ever reaching a large bet. A Paroli run can hit three wins in a row and finish with a much bigger gain than the opening amount. Fibonacci can move backwards through the sequence after a few wins.

Those runs can happen without the system having found a way around the house edge. Another set of spins can give a completely different outcome. Martingale can hit a long losing run, Paroli can repeatedly lose its larger second or third bet and Fibonacci can continue far enough for the bet to become much larger than the first unit. The reality is, a few successful sessions only show what happened during those sessions. They never change what can happen across a much bigger number of bets.

Betting Systems and Game Strategy Are Different Things

A betting system shouldn’t be confused with decisions that are part of how a casino game is played. In roulette, changing the amount after a win or loss doesn’t alter the wheel. Once the bet is placed, there is no decision that changes where the ball can land.

Blackjack is different because choices such as Hit, Stand, Double and Split are part of how the hand is played. Rules at the table can also change the house edge, including whether the dealer Hits or Stands on Soft 17 and what a natural blackjack pays.

Increasing a Blackjack bet because the previous hand lost is separate from those decisions. The larger amount doesn’t make the next cards more helpful. Casino games on PokerStars can have different rules, payouts and house edges depending on the game or version being played. A betting system just decides how much is placed from one round to the next.

Betting Systems and Modern Roulette

These betting systems have been around for centuries and are still often used when talking about roulette. Red and Black are useful examples because the bets are simple to follow. On European Roulette, Red covers 18 pockets, Black covers another 18 and zero sits outside both. That basic setup doesn’t change just because the game has moved online.

Modern roulette games on PokerStars Casino also show previous outcomes on the interface, making streaks easy to spot. Those numbers show what has already happened instead of what will happen on the next spin. If five Reds have just appeared, the next spin still starts with the same 37 pockets. Nothing has been added to the Black side of the wheel to make up for the previous spins.

Why Betting Systems Have Lasted for Centuries

Random events can still fall into patterns. A roulette wheel can land on Black six times, a coin can land Heads several times and a Blackjack player can go through a run of winning hands. Looking at those patterns afterwards can make it feel like there should be some way to use them. Betting systems give the player a rule for what to do next.

Martingale puts a doubling rule on losses. D’Alembert makes smaller changes to the bet. Fibonacci adds a famous number sequence, while Labouchère turns the process into a line that can gradually be crossed off. Having a set rule can make the betting feel less random, even when the game itself has not changed. A player can react differently to the last spin or hand, but it still has no say in what happens next.

Final Thoughts: Different Systems, Same Odds

Betting systems have been around for hundreds of years. Martingale became known for doubling after losses, Fibonacci used one of mathematics’ best-known sequences and D’Alembert increased the bet more slowly. Paroli and Labouchère took different approaches. Each system gives the player a rule for deciding the next bet. Some increase quickly, some take much longer and others only raise the amount after winning. Modern casino games on PokerStars may look very different from the tables where many of these systems were first used, but the maths hasn’t changed.

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