Atlassino Academy · Money and risk · Lesson 1 of 3
Swings, goals and running out
One evening can land far from its average cost. How far it usually lands, how long the money for the evening lasts, and what a loss limit or a point to leave ahead changes, and what it leaves alone.
An hour of 10 units bets on red at a double-zero wheel costs 20.00 units on average. How far from that does one real hour land?
The short answer
Often a long way. One hour of 38 spins at an example 10 units a spin has an average result of −20.00 units, and a usual hour lands about 61.56 units above or below that average. Play four times as long and the average cost grows four times over, while that swing only doubles. A walk-away rule changes how the evening ends; it does not change what each bet costs. The 10 units is only an example bet, not a suggested one.
Three bars on one axis, for one, two and four hours of the same example bet. Each longer evening sits further below zero, and each bar is wider; from one hour to four the average moves four times as far, the bar only twice as wide.
1 hour, 38 spins
average −20.00 units · swing 61.56 units either way
2 hours, 76 spins
average −40.00 units · swing 87.06 units either way
4 hours, 152 spins
average −80.00 units · swing 123.12 units either way
Each bar runs one swing either side of the average result: the dark mark is the average, and the dashed rule is zero. 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 dealt roulette for years, and every night someone had a good first hour and decided the wheel was being kind to them. A wheel is never kind. A first hour lands wherever the swing happens to put that hour, and the price of every spin underneath stays exactly where it was.
How far one hour lands from its price
At a double-zero wheel the house keeps 5.26% of every bet on red, on average over a very long run of spins.
One hour on this page is 38 spins, an example pace. At an example 10 units a spin, the average result of the hour is −20.00 units.
One hour: the average result, the swing either side of the average, and the range one swing below and one swing above.
The usual distance between one hour and the average is called the standard deviation; on this page it is the swing. An hour one swing below the average and an hour one swing above the average are both ordinary hours.
On one evening almost anything can happen. The average is what the same bet costs over a great many evenings, not a forecast of tonight.
Four times as long: four times the cost, twice the swing
The average cost grows in step with the number of bets, because every spin adds the same small price. The swing grows only with the square root of the number of bets: four times the spins, twice the swing.
| Evening | Spins | Money bet | Average result | Swing, either way |
|---|---|---|---|---|
| 1 hour | 38 | 380 units | −20.00 units | 61.56 units |
| 2 hours | 76 | 760 units | −40.00 units | 87.06 units |
| 4 hours | 152 | 1,520 units | −80.00 units | 123.12 units |
| 20 hours | 760 | 7,600 units | −400.00 units | 275.30 units |
From one hour to four, the average result moves four times as far below zero and the swing only twice as far. Over twenty hours the average has moved further than one swing reaches.
Over many hours, the price shows through. That is why the share of evenings still ahead keeps falling the longer the evening goes on, whatever the size of the bet.
The share of evenings that are ahead after each length of play on red at a double-zero wheel, at any bet.
Every spin starts from nothing
The wheel does not remember the spins before. A long run of one colour does not make the other colour due, and playing on does not make a good result due either.
Six spins in a row came up black, and the next spin is still to come. The plate under the row gives the chance of red on the next spin.
Red on the next spin, as on every spin: 18 in 38.
Playing longer does not raise the chance that the evening ends ahead: the arithmetic in the section above shows the share of evenings ahead falling with every hour.
How long the money lasts
Bring 100 units, bet an example 10 units on red at a double-zero wheel on every spin, and stop when a full bet is no longer left. How long that takes is not one number: the time is spread over many evenings.
Each bar is the share of evenings in one band of time, out of every way such an evening can go.
The money lasts the whole first hour in 83.84% of evenings, and the first two hours in 60.43%. By 100 spins, about 2 hours 38 minutes, half of the evenings have run out. Lasting an hour here means every bet of that hour was placed.
The band is the time until the money runs out, from the tenth to the ninetieth evening in a hundred: eight evenings in ten end inside the band, between 44 minutes and 12 hours. The solid tick is the halfway point; the dashed line is the average, about 5 hours, pulled past the halfway point by a few long evenings.
A bigger bet runs through the same money faster. With 100 units on red, half of the evenings have run out by 6 hours 51 minutes at an example 5 units a spin, and by 38 minutes at an example 25 units.
The same arithmetic works for any money, bet and game.
A goal, or running out first
Now give the same evening a goal: 100 units brought, an example 10 units a spin on red, and stop either at 200 units or when the money is gone, with no time limit.
One bar split into two parts: the evenings that reach the goal take about a quarter of the bar, and the evenings that run out first take the rest.
- Reach the goal: 25.85% of evenings.
- Run out first: 74.15% of evenings.
Red comes up on 47.37% of spins, 18 in 38, close to half. The evening as a whole reaches the goal far less often, because every spin on the way pays the price. Both shares of evenings are worked out over every way such an evening can go.
Walk-away rules: how the evening ends, not what a bet costs
A walk-away rule is a point decided before the first spin: a loss limit, a point to leave when you are ahead, a time limit, or more than one of these. The responsible-gaming pages of the provincial gaming bodies of Neuquen and Tierra del Fuego tell players to decide on a money limit before playing and to set a limit on the time they play, and the Tierra del Fuego page adds a goal for winnings fixed in advance. That is advice, not a rule of any room.
Take two hours at an example 10 units a spin, with a loss limit of 100 units and a point to leave when 100 units up.
- End at the loss limit404of 1,000 evenings
- End at the point to leave ahead141of 1,000 evenings
- End when the time is up455of 1,000 evenings
- Average result−30.68 unitsover 58.3 spins, about 583 units bet
Out of 1,000 evenings with these rules. The same two hours with no rule have an average result of −40.00 units over 76 spins.
The rules cut the average loss of the evening, and they do it by cutting the number of spins played. Here is the same evening, ended five different ways.
- ARule A: No walk-away rule: play until the time is up
- BRule B: Leave at a loss of 100 units
- CRule C: Leave when 100 units up
- DRule D: Leave at a loss of 100 units, or when 100 units up
- ERule E: Leave at a loss of 50 units, or when 50 units up
| Rule | At the loss limit | At the point to leave ahead | When the time is up |
|---|---|---|---|
| A · No walk-away rule: play until the time is up | – | – | 1,000 |
| B · Leave at a loss of 100 units | 405 | – | 595 |
| C · Leave when 100 units up | – | 141 | 859 |
| D · Leave at a loss of 100 units, or when 100 units up | 404 | 141 | 455 |
| E · Leave at a loss of 50 units, or when 50 units up | 612 | 362 | 26 |
Every rule moves evenings between the three endings.
| Rule | Average result | Spins played | Money bet | Kept by the house, of the money bet |
|---|---|---|---|---|
| A · No walk-away rule: play until the time is up | −40.00 units | 76 | 760 units | 5.26% |
| B · Leave at a loss of 100 units | −33.09 units | 62.9 | 629 units | 5.26% |
| C · Leave when 100 units up | −37.59 units | 71.4 | 714 units | 5.26% |
| D · Leave at a loss of 100 units, or when 100 units up | −30.68 units | 58.3 | 583 units | 5.26% |
| E · Leave at a loss of 50 units, or when 50 units up | −12.61 units | 24 | 240 units | 5.26% |
Every rule changes how many spins get played, and with them the money bet and the average result. The last column does not move.
The share of the money bet that the house keeps, under each rule above: the same bar every time.
The bars show it by calculation, and it holds for every loss limit, every point to leave ahead and every time limit, because every spin carries the same price whatever came before the spin.
Mayaformer floor supervisor · manners, money and breaks
A good first hour is swing, not a trend. Write the walk-away rule down before you go, and keep the rule when the moment comes, whichever way the evening has gone.
Money already lost is the price of the evening, not money the table owes you. If keeping the rule starts to feel impossible, the lines below are there for exactly 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.
Show all provinces
If you are thinking about suicide, call 135 or 0800 345 1435 from anywhere in the country
Practice: how long the money lasts
Level 1 sets 100 units on one game at an example bet, and you say by when the money has run out in half of the evenings. The games are red or black at a double-zero wheel and at a single-zero wheel, and the pass line at craps. Level 2 sets one to four hours on red at an example bet the plan chooses, and you pick among walk-away rules.
Nothing is scored here. Both levels are read through. Every answer shows the exact figures first and, in grey, one run of a thousand evenings beside them.
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Level 1 · How long the money lasts
THE EVENING
- GAMEred or black at a double-zero wheel
- EXAMPLE BET10 units on every spin
- MONEY FOR THE EVENING100 units
- PACE38 spins an hour
The evening stops when a full bet is no longer left.
100 units for the evening, on red or black at a double-zero wheel, at an example bet of 10 units on every spin. Picture many evenings like this one. By when has the money run out in half of these evenings?
- Less than 1 hour 16.16% of evenings ran out inside this band. By the end of this band fewer than half of the evenings had run out: the halfway point comes later, at 2 hours 38 minutes.
- 1 to 2 hours 23.42% of evenings ran out inside this band. By the end of this band fewer than half of the evenings had run out: the halfway point comes later, at 2 hours 38 minutes.
- 2 to 4 hours Half of the evenings had run out by 2 hours 38 minutes, after 100 spins, and that falls in this band. 24.46% of evenings ran out inside this band.
- 4 hours or more This is where the average falls, about 5 hours: a few very long evenings pull the average past the halfway point. Half of the evenings had already run out by 2 hours 38 minutes. 35.97% of evenings lasted at least 4 hours.
The money lasted the whole first hour in 83.84% of evenings and the first two hours in 60.43%. Eight evenings in ten ran out between 44 minutes and 12 hours; half had run out by 2 hours 38 minutes, and the average is about 5 hours. One run of 1,000 evenings at this table: the money lasted about 4 hours 51 minutes on average, and eight evenings in ten ran out between 44 minutes and 11 hours. A run of 1,000 evenings is a sample: its average wanders around the exact average, which is worked out separately and does not change from run to run.
Level 2 · Walk-away rules
THE PLAN
- GAMEred or black at a double-zero wheel
- EXAMPLE BET10 units a spin, set by the plan
- TIME2 hours, 76 spins
WALK-AWAY RULE: still to be written
2 hours on red at a double-zero wheel, at an example 10 units a spin that the plan sets. Under which of these walk-away rules do the most evenings end at the loss limit?
- Leave at a loss of 100 units, or when 100 units up The loss limit ends fewer evenings under this rule than under the answer. Out of 1,000 evenings, 404 end at the loss limit, 141 end at the point to leave ahead and 455 when the time is up. The average result is −30.68 units over 58.3 spins, about 583 units bet. The house kept 5.26% of that money.One run of 1,000 evenings, on the same spins for every rule here: 391 end at the loss limit, 157 end at the point to leave ahead and 452 when the time is up; average result −27.88 units.
- Leave at a loss of 100 units The loss limit ends fewer evenings under this rule than under the answer. Out of 1,000 evenings, 405 end at the loss limit and 595 when the time is up. The average result is −33.09 units over 62.9 spins, about 629 units bet. The house kept 5.26% of that money.One run of 1,000 evenings, on the same spins for every rule here: 392 end at the loss limit and 608 when the time is up; average result −30.86 units.
- Leave at a loss of 50 units, or when 100 units up The loss limit ends the most evenings under this rule. Out of 1,000 evenings, 706 end at the loss limit, 129 end at the point to leave ahead and 165 when the time is up. The average result is −19.31 units over 36.7 spins, about 367 units bet. The house kept 5.26% of that money.One run of 1,000 evenings, on the same spins for every rule here: 708 end at the loss limit, 145 end at the point to leave ahead and 147 when the time is up; average result −18.52 units.
A loss limit that sits closer, or a point to leave ahead that sits further off, sends more evenings to the loss limit. Under every rule the house keeps 5.26% of the money bet. A run of 1,000 evenings is a sample: its average wanders around the exact average, which is worked out separately and does not change from run to run.
Every evening in the practice run is worked out exactly from the outcomes of one round of the game, and every grey line beside an exact figure is one run of a thousand evenings from a fixed starting point.
How the practice run works
Level 1. The time until the money runs out is worked out exactly, round by round, from the outcomes of one round of the game; an evening stops when a full bet is no longer left. An evening lasted an hour when every bet of that hour was placed. The band is the time until the money runs out, from the tenth to the ninetieth evening in a hundred.
Level 2. Every rule is worked out exactly over every way the spins can go. An evening ends at the loss limit or at the point to leave ahead the first time the result reaches either one, and otherwise when the last spin of the time has been played; an evening that reaches the loss limit or the point to leave ahead on that last spin counts as ending there, not as ending when the time is up.
The grey lines. Each grey line is one run of a thousand evenings from a fixed starting point, the same spins for every rule in a level 2 question, so the rules are compared on the same luck. A run is a sample, and the exact figure above each grey line is the one to read.
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Check yourself
Mayaformer floor supervisor · manners, money and breaks
Six quick ones. Pick an answer, then check it. No timer, no points: just whether it stuck.
Back to the question we opened with. One hour at an example 10 units a spin on red lands about 61.56 units from its average either way, and the average of −20.00 units is still the price of every hour, whichever way that hour lands.
Where to go next, and further reading
18+ under each province’s rules · Play responsibly · the helplines are further up this page
Further reading
- 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.
How it’s worked out
Numbers, pace, sources and hosts
Where this happens. Every evening on this page is at a live table in a room you can walk into: red or black at a roulette wheel, and in the practice run the pass line at craps. Electronic tables and screens at home keep a pace of their own and are not counted here.
Numbers. Every figure on this page is worked out when the page is built from the outcomes of one round of each game, counted exactly: the average result and the swing of an evening, the share of evenings ahead, the chance of reaching a goal, the time until the money runs out, and every walk-away rule. The swing is one standard deviation of the result of the evening; no operator material we could read uses that term, and the figures for the swing are our own calculation. No document we could read gives the chance of reaching a goal like the one here, or says that a walk-away rule leaves the price of each bet alone: both are our own calculation as well. The only figures from a sample are the grey lines beside the exact ones. Our checks recompute every figure from the engine rather than reading it back from this page.
Pace. No Argentine document we could read prints how many rounds a table plays in an hour. Roulette uses 38 spins an hour at a double-zero wheel and 35 at a single-zero wheel, example paces rather than measured ones. Craps uses about 60 rolls an hour, the pace casino rating systems are set to, as a casino consultant describes them in Global Gaming Business, which comes to about 18 pass-line decisions. The figures carry no source button on this page, and a real table can be faster or slower.
What is deliberately absent. There is no betting system on this page: doubling after a loss and the table maximum are lesson 10.2, which leans on the walk-away arithmetic here instead of repeating it. Chasing a loss is lesson 10.3. Setting the limits of an evening is lesson 0.2; runs of one colour and hot numbers are lessons 2.8 and 3.17; helplines and self-exclusion are guides of their own. Why each bet carries the edge it carries belongs to the lessons of each game. Blackjack is not in the practice run, because the outcomes of a whole blackjack hand come from a published table we do not quote. Baccarat is not in the practice run either: the punto y banca of the provincial rulebooks takes its commission in its own way, which is lesson 6.8. No bet sizes are compared for reaching a goal: every figure is for the example bet named beside it, and none of them says which bet to make.
Sources. Every claim from a document 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. Every game on this page is played against the house. Knowing how far an evening swings and what a walk-away rule does lets you plan the evening; nothing on this page makes any bet cheaper, and nothing here says how an evening would end.
One evening is the short run
The amount actually returned to players in any one hour or day, or even week — and therefore the cost of play — will differ from this significantly.
Productivity Commission (Australian Government) · Gambling, Inquiry Report no. 50, volume 1 (2010) · read
The price adds up the longer you play
As a rule of thumb, drift dominates swings in the long term.
Walk-away rules as operators print them
Antes de empezar, siempre establecer un límite de dinero.Free translation: Before starting, always set a limit on money.
Asegurarse de establecer un período de tiempo limitado.Free translation: Make sure to set a limited period of time.
Decida de antemano lo que desea gastar durante su sesión de juego.Free translation: Decide in advance what you wish to spend during your gaming session.
PONERSE UN TIEMPO LIMITE PARA JUGAR Y RESPETARLO.Free translation: SET A TIME LIMIT FOR PLAYING AND RESPECT IT.
Fije un objetivo predeterminado para sus ganancias, guárdese algo, y apueste solo la cantidad restante.Free translation: Set a predetermined target for your winnings, keep something aside, and bet only the remaining amount.
Instituto Provincial de Regulación de Apuestas (IPRA) · IPRA — Programa Juego Responsable · read
One evening is the short run
The amount actually returned to players in any one hour or day, or even week — and therefore the cost of play — will differ from this significantly.
Productivity Commission (Australian Government) · Gambling, Inquiry Report no. 50, volume 1 (2010) · read
The price adds up the longer you play
As a rule of thumb, drift dominates swings in the long term.
Each round is independent
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.
Playing on does not make a result due
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.
Walk-away rules as operators print them
Antes de empezar, siempre establecer un límite de dinero.Free translation: Before starting, always set a limit on money.
Asegurarse de establecer un período de tiempo limitado.Free translation: Make sure to set a limited period of time.
Decida de antemano lo que desea gastar durante su sesión de juego.Free translation: Decide in advance what you wish to spend during your gaming session.
PONERSE UN TIEMPO LIMITE PARA JUGAR Y RESPETARLO.Free translation: SET A TIME LIMIT FOR PLAYING AND RESPECT IT.
Fije un objetivo predeterminado para sus ganancias, guárdese algo, y apueste solo la cantidad restante.Free translation: Set a predetermined target for your winnings, keep something aside, and bet only the remaining amount.
Instituto Provincial de Regulación de Apuestas (IPRA) · IPRA — Programa Juego Responsable · read