Atlassino Academy · Money and risk · Lesson 2 of 3
The Martingale: doubling after a loss
Doubling the bet after every loss ends almost every series a little up. Here is the rare series that takes back what all the others brought in, how the table maximum ends the doubling, and why each unit bet keeps its price.
Say a player starts with 5 units on red and doubles the bet after every loss until a bet is paid. That run of bets is one series, and almost every series ends 5 units up. Does doubling make roulette cheaper?
The short answer
No. At an example table with a zero and a double zero, a minimum of 5 units and a maximum of 500 units, about 989 series in 1,000 do end 5 units up. The other 11 lose all 7 bets the doubling can make under the table maximum, one after another, and each of those series costs 635 units. Averaged over every series, the rare big loss outweighs the many small ups, and each unit bet costs what a single bet on red costs. The doubling changes how the results are spread, not the price of each unit bet.
Two bars for the series at the example table: a bar almost the full width for the series that end with the first bet up, and a sliver for the series that lose every bet of the ladder.
End 5 units up
about 989 of 1,000 series.
Lose every bet of the ladder, 635 units
about 11 of 1,000 series.
The same example table in both bars. The length of each bar is how many series in 1,000 end that way, worked out exactly rather than played. Every amount on this page is only an example, not a suggested bet. Amounts are in betting units, not pesos: a unit can stand for any amount, and the arithmetic is the same at any scale.
Leoformer dealer · rules and maths
I used to watch people at the wheel doubling the bet after every spin that did not come up their colour. Almost every run of doubled bets ended a little up. Every so often one ended with nothing left on the table in front of the player.
Nobody at the table ever saw those two kinds of run side by side. Side by side is the only way to see what the doubling costs, so that is how this lesson lays them out.
Double the bet after every loss
The Martingale is a simple betting system: double the bet after every loss, and go back to the first bet once a bet is paid. Some people call this doubling down. This page says doubling the bet, because doubling down is a different move at the blackjack table.
Every series on this page is a bet on red. A bet on red is paid 1 to 1 when the ball comes to rest in one of the red pockets, 18 in 38 on a wheel with a zero and a double zero.
- First bet5 unitson red, lost: the series is 5 units down.
- Second bet, doubled10 unitson red, lost: the series is 15 units down.
- Third bet, doubled20 unitson red, paid 20 units: the series ends 5 units up.
A series that loses twice and is paid on the third bet. The bets a series can ask for, each twice the bet before, are its ladder. Each bet in the ladder equals all the bets before it added together, plus the first bet, so a paid bet always leaves the series up by exactly the first bet.
That is the appeal of the doubling: whichever bet is finally paid, the series ends one first bet up. The catch is the word finally. The doubling needs a paid bet before the bets outgrow the table or the money.
The table maximum ends the doubling
A live table has a smallest and a largest bet. In Buenos Aires Province, the standards for casinos require the minimum and maximum bet of every live table game to be posted where the players at the table can see them. What that maximum does to a doubling is plain arithmetic.
Bars for the bets of one series at the example table, each bar twice the height of the bar before, with a dashed line across at the table maximum. The last bar, the next bet the doubling would ask for, rises above the line.
- 5 units
- 10 units
- 20 units
- 40 units
- 80 units
- 160 units
- 320 units
- 640 units
The dashed line is the table maximum, 500 units.
The bets of one series at the example table, each twice the bet before it. The last bar is the next bet the doubling would ask for, and that bet is over the dashed line, so the table will not take it.
After 7 losses in a row the series is 635 units down, and the next bet the doubling needs, 640 units, is over the maximum, so the table will not take it. The series is over at a loss, and a new series starts again from 5 units. A bet at the maximum instead does not rescue the series: paid in full, a bet at the maximum still leaves the series 135 units down.
Many small ups, one rare big loss
Seen one series at a time, the doubling looks safe, because the big loss is rare. Seen over 1,000 series, the small ups and the rare big losses can be set against each other.
Two bars over many series at the example table: the money the series that end up bring in, and the longer bar for the money the full ladders take out.
Brought in by the series that end up
about 4,944 units over 1,000 series.
Taken by the series that lose every bet
about 7,104 units over 1,000 series.
Over 1,000 series at the example table, on average: the money the small ups bring in, and the money the rare full ladders take out. The second bar is longer, and the difference between the two bars, shared out over the series, is the average cost of one series.
- One series, on average2.16 unitslost.
- Money bet in one series, on average41.04 unitsin all the bets of the series.
Divide the average loss of a series by the money bet in an average series, and the answer is 5.26% of every unit bet. A single bet on red at this wheel carries exactly the same share: 5.26%.
The doubling is a rule for how much to bet, and it leaves that share where the wheel puts it. Rules for when to stop leave it there too; lesson 10.1 shows why.
Another table, the same doubling
Two things on a table sign can change: the maximum and the wheel. Here is the same doubling from 5 units on red at a table with a higher maximum, and at a wheel with one zero.
| Example table | Bets in the ladder | The next bet, over the maximum | A series loses every bet | That series costs |
|---|---|---|---|---|
| Two zeros, 5 units to 500 units | 7 | 640 units | about 1 in 89 | 635 units |
| Two zeros, 5 units to 1,000 units | 8 | 1,280 units | about 1 in 170 | 1,275 units |
| One zero, 5 units to 500 units | 7 | 640 units | about 1 in 106 | 635 units |
- Two zeros, the lower maximum5.26%of every unit bet, with the doubling.The same bet on red at every spin, with no doubling, keeps 5.26%.
- Two zeros, the higher maximum5.26%of every unit bet, with the doubling.The same bet on red at every spin, with no doubling, keeps 5.26%.
- One zero2.70%of every unit bet, with the doubling.The same bet on red at every spin, with no doubling, keeps 2.70%.
All three are example tables, not rooms. Another table changes how rarely the full ladder is lost and what it costs, not the share of each unit bet: that share is the edge of the wheel, the same for the doubling and for a flat bet, the same bet on every spin.
The maximum of a real table is on the sign beside it; every table on this page is an example.
The money runs out first
An evening has a second limit besides the table maximum: the money brought for the evening. Take 100 units, an example amount for the evening and not a suggested one, and the same doubling from 5 units.
Bars for how far down the series is after each loss in a row, rising faster and faster, with a dashed line across at the money for the evening. The last bar rises above the line.
- 5 units
- 15 units
- 35 units
- 75 units
- 155 units
The dashed line is the money for the evening, 100 units.
Each bar is how far down the series is after that many losses in a row; the last bar is where the next doubled bet would take the series. After 4 losses only 25 units is left, and the next bet the doubling needs is 80 units: the money ends the doubling long before the table maximum does.
Doubling spends a budget much faster than a flat bet does, and a run of losses can take all of the money quickly. Under the doubling sits the idea that a paid bet must come soon, and the wheel owes nothing: a run of losses makes the next spin no more likely to be red.
The Martingale is one of the strategies sold as a way round the house, and the Martingale is a myth. No order of bets gets round the house edge, because each bet carries the edge whatever came before it.
Mayaformer floor supervisor · manners, money and breaks
A system that ‘always comes out a little ahead’ is hiding a rare big loss. Set at home what the evening may cost, and let that amount end the evening, not the next bet the doubling asks for.
Doubling to get back what a full ladder took is chasing a loss, and lesson 10.3 is about exactly that. If the money on the table starts to feel like money you need, the lines below are there for that.
If it stops being fun
If gambling stops being entertainment, help is available. Minimum entry age: each province sets its own. Where we found the rule or regulator that sets it, we cite it on the page; those we found set it at 18. More on responsible gaming.
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If you are thinking about suicide, call 135 or 0800 345 1435 from anywhere in the country
Practice: Martingale meets the table limit
Each table in the practice run is the sign of an example table: the wheel, the minimum bet and the maximum bet. The doubling starts at the minimum on red, and the question is how its series end: how long the ladder is, what a full ladder costs, how often a series loses every bet, and what share of the money stays with the house. Level 2 changes the table: a higher maximum, then a wheel with one zero.
Nothing is scored here. Both levels are read through in order, and every answer is worked out exactly from the ladder of bets and the chance of red. Nothing spins: each series is priced, not played.
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Level 1 · The example table
Table sign
- WheelZero and double zero
- Minimum bet5 units
- Maximum bet500 units
An example table, not a room
First bet of the doubling: 5 units on red
Every bet of the ladder loses, one after another, up to the last bet the table will take. What has the series cost?
- 320 units That is only the last bet of the ladder. Every bet before the last bet was lost as well.
- 635 units Every bet of the ladder added together: 5 units, 10 units, 20 units, 40 units, 80 units, 160 units, 320 units.
- 35 units That is 7 bets of 5 units each. In a doubling each bet is twice the bet before it.
- 640 units That is the next bet the doubling would ask for, and the table does not take a bet over its 500 units maximum.
A full ladder costs every bet in the ladder added together, which comes to just under the next bet the doubling would ask for.
Level 2 · Other tables
Table sign
- WheelOne zero
- Minimum bet5 units
- Maximum bet500 units
An example table, not a room
First bet of the doubling: 5 units on red
At this table the house keeps 2.70% of every unit bet, against 5.26% at the wheel with two zeros. Why?
- The doubling works better on this wheel The doubling is the same at both tables. Bet the same on red at every spin with no doubling, and this wheel keeps the same smaller share.
- The wheel has one zero instead of two Every bet on red at this wheel keeps the smaller share, with the doubling or without the doubling.
- The table maximum is easier to reach The maximum is the same 500 units at both tables, and so is the ladder.
The smaller share belongs to the wheel, not to the doubling: the doubling decides how much each bet is, and the wheel decides what each unit bet costs.
Every table in the practice run is worked out the same way: the ladder of bets the doubling asks for between the minimum and the maximum, and the chance that one spin comes up red at that wheel.
How the practice run works
Where the figures come from. Every figure is worked out when the page is built: the ladder of bets between the minimum and the maximum, and the chance that one spin comes up red at that wheel. A count such as “about 989 of 1,000 series” is the exact share rounded to a count, not a count from a run of series. Every wrong answer is one named mistake: the next bet counted into the ladder, the minimum divided into the maximum, the last bet taken for the whole ladder, the share of series taken for the share of money, the edge of the wheel credited to the doubling.
Why there are no squares to fill. Both levels are for reading: there is nothing to count, so there is no strip, and every answer is explained.
Your progress stays in this browser, and nothing is sent anywhere.
Check yourself
Mayaformer floor supervisor · manners, money and breaks
Three quick ones. Pick an answer, then check it. No timer, no points: just whether it stuck.
Back to the question we opened with. Doubling does not make roulette cheaper. The small ups are real, the rare full ladder takes back more than they bring in, and the house keeps 5.26% of every unit bet, as it does from a single bet on red.
Where to go next, and further reading
18+ under each province’s rules · Play responsibly · the helplines are further up this page
Further reading
- Introduction to Probability (open version) · Charles Grinstead and Laurie Snell, Dartmouth CollegeThe roulette example worked out, and why no system of play changes who the game favours.
- Lecture 26: Random walks, the Gambler’s Ruin · MIT 6.042J Mathematics for Computer Science (2000)Why even a very large bankroll rarely reaches a small target on a roulette wheel.
- Teaching a university course on the mathematics of gambling · Stewart Ethier and Fred Hoppe, arXiv (2020)How a university course explains the house advantage and betting systems such as the martingale.
- The Gambler’s Fallacy and the Hot Hand: Empirical Data from Casinos · Rachel Croson and James Sundali, Journal of Risk and Uncertainty (2005)What gamblers at a real roulette table believed about “due” numbers, measured from casino video.
How it’s worked out
Tables, numbers, sources and hosts
Where this happens. Every table on this page is a live roulette table in a room you can walk into, and every series is a run of bets on red. Every table is an example, named by its wheel, its minimum and its maximum, and none is the table of any particular room.
Numbers. Every figure is worked out when the page is built, exactly, from two things: the ladder of bets the doubling asks for between a minimum and a maximum, and the chance that one spin comes up red at that wheel. A series is one ladder, from the first bet to the first paid bet or to the end of the ladder. A count of series out of 1,000 is the exact share rounded to a count, not a count from a run. The average loss and the average money bet in a series are exact as well, and our checks recompute every figure from the engine rather than reading the figure back from this page.
What is deliberately absent. No other betting system is named or described here: none of the documents we could read prints the rules of another system, and what is said of systems in general is said in the words of the guides behind the source buttons. Why no rule for when to stop changes the price of a unit bet is lesson 10.1, together with how far an evening swings and how long money lasts; for the doubling, a rule for how much to bet, this page works it out at three example tables. Chasing a loss is lesson 10.3. Why every bet on one wheel costs the same is lesson 4.3. Setting a limit for the evening is lesson 0.2. There is no formula here for sizing bets, and none anywhere in the course. No document we could read says that a table maximum stops a doubling: the page shows it by arithmetic. The maximums of real tables are not on this page; the rooms we list publish them.
Sources. Every claim on this page carries a button with the sentence it rests on, the publisher who wrote that sentence and the date we read it.
Hosts. Leo and Maya are illustrated characters. Their backstories, a former dealer and a former floor supervisor, are written for the course: neither is a real person, and neither speaks for any casino. The rules and numbers on this page come from the sources it cites, not from their experience.
No promise. Roulette is played against the house. Knowing the shape of a doubling lets you expect the rare big loss; nothing on this page makes any bet cheaper, and nothing here says how an evening at the wheel would end.
Doubling and the money for the evening
or in the unlikely event that black turns up six times in a row so that you are down 63 dollars and cannot make the required 64-dollar bet.
That is, if the gambler starts with a trillion dollars, then his chance of going up a mere $100 before losing the whole trillion is still less than 1 in 37,648!
The edge on a double-zero wheel
−1×37/38 + 35×1/38 = −0.0526 (5.26% de margen de la casa)Free translation: −1×37/38 + 35×1/38 = −0.0526 (5.26% house edge)
The 0 and 00 slots are green, and half of the remaining 36 slots are red and half are black.
If you bet 1 dollar on red, you win 1 dollar if the ball stops in a red slot, and otherwise you lose a dollar.
E(X) ≈ −.0526The Martingale as a myth
Unfortunately, such systems cannot change even a fair game into a favorable game.
the martingale system, or any other system for that matter, cannot make the game into a favorable game.
What the Martingale is
In the martingale system, one doubles one’s bet after each loss until finally achieving a win.
Every time you win, bet 1 dollar next time. Every time you lose, double your previous bet.
Recall that in the martingale doubling system (see Exercise 1.1.10), the player doubles his bet each time he loses.
Live table limits are posted
El público podrá tomar conocimiento de los valores de las fichas, mínimos y máximos en vigencia, exhibidos en la sala de juego, dándose como conocidos por el Apostador en el momento de efectuar su apuesta.Free translation: The public may find out the values of the chips and the minimums and maximums in force, displayed in the gaming room, and these are deemed known by the Bettor at the moment of placing their bet.
Section Capítulo 1, Ruleta Americana: Reglamento del Juego (1.1 a 1.10)
The edge on a double-zero wheel
−1×37/38 + 35×1/38 = −0.0526 (5.26% de margen de la casa)Free translation: −1×37/38 + 35×1/38 = −0.0526 (5.26% house edge)
The 0 and 00 slots are green, and half of the remaining 36 slots are red and half are black.
If you bet 1 dollar on red, you win 1 dollar if the ball stops in a red slot, and otherwise you lose a dollar.
E(X) ≈ −.0526Doubling and the money for the evening
or in the unlikely event that black turns up six times in a row so that you are down 63 dollars and cannot make the required 64-dollar bet.
That is, if the gambler starts with a trillion dollars, then his chance of going up a mere $100 before losing the whole trillion is still less than 1 in 37,648!
Playing on and the next spin
For example, the Gambler’s Ruin random walk is a martingale because the outcome of the current bet is independent of the outcomes of all preceding bets.
The gambler’s fallacy is a belief in negative autocorrelation of a non-autocorrelated random sequence.
Thus he believes a tail is “due,” that is, more likely to appear on the next flip than a head.
The Martingale as a myth
Unfortunately, such systems cannot change even a fair game into a favorable game.
the martingale system, or any other system for that matter, cannot make the game into a favorable game.
Systems and the house edge
If you are in a casino, you will see players adopting elaborate systems of play to try to make unfavorable games favorable.
Unfortunately, such systems cannot change even a fair game into a favorable game.
As a rule of thumb, drift dominates swings in the long term.