Atlassino Academy · How casinos work · Lesson 1
The house edge: a price inside every bet
Every bet carries a price that is settled before the ball drops, and you can find that price by counting pockets: here is where the price hides in the payout, and how to count it out.
Two bets on the same wheel, the same money on each: one chip on a single number, one chip on a dozen. The single number is paid rarely and pays a lot; the dozen is paid more often and pays a little. Which of the two bets costs you more?
The short answer
A wheel with two zeros has 38 pockets, and the table pays a single number 35 to 1. A payout of 37 to 1 would leave nobody ahead after one full turn of the wheel: one chip lost on each of the 37 pockets that are not yours, 37 chips paid on the one pocket that is. The table pays two chips less than that, and those two chips out of 38 pockets are the house edge (the share of all the money bet that the casino keeps on average, over many rounds) — here, 5.26%.
Count the pockets on the wheel and the pockets your bet covers; the payout that would leave nobody ahead follows from those two counts. Whatever the table pays instead leaves some pockets out, and those pockets, as a share of the wheel, are the price of the bet.
Leoformer dealer · rules and maths
I dealt for years, and the question I was asked most was some version of “which of these is the good bet?” Nobody expects the answer, which is that the table says the price out loud before a chip goes down: the price of every bet is in its payout.
Once you can read the price off the payout, you never need anybody’s opinion about a bet again.
A price, decided before the ball drops
You know what a coffee costs before you order it. A bet looks like the one purchase that works the other way round: you hand the money over first and find out afterwards what the bet cost you. It is not. The price of a bet is settled before the ball is thrown, and the only thing the spin decides is which side pays that price tonight.
The price is an average. It is what a bet loses over many spins, measured against the money you put down on it. Three things follow from that, and none of the three is obvious.
- The price is an average, not tonight’s result.You can sit down, be paid twice and walk out ahead, and the price was still charged on every bet you made. The price shows in the average over many bets, never in any one evening.
- The price is charged on each bet, not on your wallet.The same chips go down again and again across an evening, and the price is charged every time they go down.
Leoformer dealer · rules and maths
People hear “the house has an edge” and take it as a mood, as if the room were simply against them. The house edge is a number. It is the same number for a first-timer and for somebody who has played for thirty years, and you can work it out in ten seconds once you know where to look.
Where the price hides: count the pockets
Here is a wheel with two zeros: 38 pockets, the numbers 1 to 36, a zero and a double zero. Your chip is on 17.
Your chip covers 1 pocket of 38
000123456789101112131415161718192021222324252627282930313233343536The marked pocket is your number. The other 37 pockets belong to the table.
Now count. Picture one full turn of the wheel: 38 spins, with the ball landing in each pocket once. On 37 of those spins your number stays away and you lose one chip each time. On the one spin your number lands, the table pays you. If the table paid 37 chips for that one chip, you would end the full turn exactly where you started, 37 chips out and 37 chips in. That payout has a name, the level payout, and on a single number the level payout is 37 to 1.
Two chips out of 38 is the price of a single number. Written as a share of the wheel, two pockets out of 38 comes to 5.26%, and that share is what the house edge of the bet means.
The price of a bet is the share of the wheel that its payout leaves out. The price is not the size of the payout, it is not how often the bet comes in, and it is not how the last hour went.
The same count in money. Put 10 units on that number and keep it there for one full turn of the wheel, and you have made 38 bets of 10 units, one on each spin.
- Goes down over the full turn380 units38 bets of 10 units, one bet on each spin
- Comes back360 unitsthe one spin your number lands, paid 35 to 1, with the bet it was on handed back
- Stays with the table20 unitsone bet for each of the 2 pockets the payout leaves out
Amounts are in betting units, not pesos: a unit can stand for any amount, and the arithmetic is the same at any scale.
One wheel, one price
Now do the same count for a dozen. A dozen covers 12 of the 38 pockets, and the table pays it 2 to 1: two chips paid for each chip staked.
A dozen covers 12 pockets of 38
000123456789101112131415161718192021222324252627282930313233343536The 12 marked pockets are the first dozen, 1 to 12. The other 26 pockets, the zero and the double zero among them, belong to the table.
| The count | A single number | A dozen |
|---|---|---|
| Pockets that are yours | 1 | 12 |
| What the table pays | 35 to 1 | 2 to 1 |
| Chips in your hand after a spin the bet is paid on, for one chip staked | 36 | 3 |
| Chips back over one full turn, for 38 chips put down | 36 | 36 |
| Pockets the payout leaves out | 2 | 2 |
| The level payout | 37 to 1 | 13 to 6 |
| The price of the bet | 5.26% | 5.26% |
That is not a coincidence, and it is the heart of the whole lesson: the payouts on this cloth are built for a wheel of thirty-six pockets. A single number paid 35 to 1 brings back 36 chips over a full turn. A dozen paid 2 to 1 brings back 36. A wheel with thirty-six pockets and no zeros would take exactly 36 chips over a full turn, one on each spin, and both of those payouts would be a level payout on it. This wheel has 38 pockets, and the two extra pockets, the zero and the double zero, are the two chips that those payouts leave out.
| Bet | Pockets it covers | How often it is paid | What the table pays | The level payout | The price |
|---|---|---|---|---|---|
| One number | 1 | 2.63% | 35 to 1 | 37 to 1 | 5.26% |
| Two numbers | 2 | 5.26% | 17 to 1 | 18 to 1 | 5.26% |
| Three numbers | 3 | 7.89% | 11 to 1 | 35 to 3 | 5.26% |
| Four numbers | 4 | 10.53% | 8 to 1 | 17 to 2 | 5.26% |
| Zero, double zero, 1, 2, 3 | 5 | 13.16% | 6 to 1 | 33 to 5 | 7.89% |
| Six numbers | 6 | 15.79% | 5 to 1 | 16 to 3 | 5.26% |
| A dozen | 12 | 31.58% | 2 to 1 | 13 to 6 | 5.26% |
One row is not like the others. The bet on the zero, the double zero and 1, 2 and 3 covers five pockets and is paid 6 to 1, and 6 to 1 on five pockets brings back only 35 chips over a full turn, not 36. That payout leaves 3 pockets out instead of 2, and 3 pockets of 38 is 7.89%.
Red, black, odd, even, high and low are not in the table. When the ball comes to rest in a zero, these bets lose only half of the chip, and the other half comes back. Over a full turn, red paid 1 to 1 brings back 36 chips on its 18 pockets and half a chip on each of the two zeros, so the payout leaves only 1 chip of the 38 out: 2.63%, half the price of a single number. That price is our count from the rule; the rule itself prints no price. The 5.26% that the university guide behind the source buttons gives for every bet on this wheel but one is the price on a wheel where a zero takes the whole chip, as it does here on the numbers and the dozens.
The same payout on a shorter wheel
Take the double zero off the wheel and change nothing else. A single number is still paid 35 to 1, and 35 to 1 still brings back 36 chips over a full turn. But a full turn of this wheel is 37 spins, not 38, so the payout leaves only one pocket out instead of two.
| A single number | Wheel with a zero and a double zero | Wheel with one zero |
|---|---|---|
| Pockets on the wheel | 38 | 37 |
| What the table pays | 35 to 1 | 35 to 1 |
| The level payout | 37 to 1 | 36 to 1 |
| Pockets the payout leaves out | 2 | 1 |
| The price of the bet | 5.26% | 2.70% |
The price is about half, on a wheel that looks the same as the other one from a metre away. The payout did not change; the number of pockets that the payout had to stretch across did.
Count the zeros before you sit down
You can check the number of pockets with your own eyes, and you do not need to count all of them: count the zeros. The national catalogue of games describes the wheels it allows in its own words:
En el extremo de la mesa se encuentra el cilindro o rueda, en el cual se ubican 37 casillas, enumeradas del 1 al 36, además de un cero.Free translation: At the end of the table is the cylinder or wheel, which has 37 pockets, numbered from 1 to 36, plus a zero.
La Ruleta Americana con un cero significa que tiene, entre las suertes o posibilidades, el signo de un cero.Free translation: American Roulette with one zero means that, among its outcomes or possibilities, it has the sign of a zero.
La Ruleta Americana con doble cero significa que tiene, entre las suertes o posibilidades, el signo de un cero y además un doble cero.Free translation: American Roulette with double zero means that, among its outcomes or possibilities, it has the sign of one zero and also a double zero.
- Look at the wheel. One zero, or a zero and a double zero?
- Look at the head of the betting cloth. The zeros on the wheel are printed on the layout as well, so a wheel you cannot see past other players can still be read from the cloth.
- Two zeros means 38 pockets, and about twice the price of one zero on every bet the two wheels share. One zero means 37 pockets.
That check takes thirty seconds, standing up, before any money goes down.
House edge, return to player, hold
House edge and return to player are one measurement seen from its two ends. Hold is a different measurement, and it is often mistaken for the house edge.
| Name | What it measures | On a wheel with two zeros |
|---|---|---|
| House edge | The share of each unit bet that stays with the table, on average, over many bets. | 5.26% |
| Return to player | The same figure from the other end: the share of the money bet that comes back. House edge and return to player add up to the whole unit. | 94.74% |
| Hold | A different measurement: counted from the money you brought to the table, not from each bet. Because the same money is bet many times over, hold comes out far bigger than the house edge. | no single figure: it depends on how many times the money goes down |
On a wheel, house edge and return to player are one subtraction apart: when 5.26% of the money bet stays with the table, 94.74% of it comes back, and on the wheel with one zero 2.70% stays and 97.30% comes back. Slot machines do not print their return to player at all; lesson 2.2 explains why, and what can be read on a machine instead.
Hold is the name to watch. Hold counts from the money you brought to the table, not from each bet, and because the same money goes down over and over, hold comes out far larger than the house edge. That is why an evening costs more than the price of one bet suggests.
Practice: set a level payout
You are not choosing a bet here, and there is nothing to guess. The wheel is in front of you, your chip is already down, and the only question is what that bet would have to pay to cost nothing at all. Count the pockets that are yours, count the pockets that are not, and the level payout falls out of the two counts.
The practice is one run through three levels. Level 1 is a single number or a pair of numbers, on either wheel. Level 2 is the rest of the cloth: dozens, corners, lines and the one bet with a price of its own. Level 3 turns the question round: the payout is given, and you count the price. Every square on the strip is one moment of the run, and all of the squares are there from the start; when the squares of one level are full, the next level begins by itself.
Lesson done
Review my mistakes (0)
Progress stays in this browser; nothing is sent anywhere.
The practice table needs JavaScript. Here is one moment from it, with the answer.
Level 1 · One number
A wheel of 38 pockets
38 pockets · your bet covers 1
000123456789101112131415161718192021222324252627282930313233343536Your number: 17. The table pays this bet 35 to 1.
What would this bet have to pay to cost nothing at all?
The answer: 37 to 1.
Count, do not guess. 1 pocket is yours and 37 are not, so the level payout is 37 to 1. The table pays 35 to 1 instead, and the 2 pockets between the two payouts are the price: 5.26% of every unit bet.
Every moment is worked out the same way: count the pockets your bet covers, count the pockets it does not cover, and the payout that balances the two counts is the level payout. What the table pays instead of the level payout is the price.
How the practice table works
Where the moments come from. Nothing is written down as an answer anywhere. Each moment names a wheel and a bet, and the count is done when the page is built: pockets covered, pockets not covered, the payout that balances the two. The wheels and what their tables pay are one model, the same one every figure on this page is counted with, so a figure in the practice and a figure in the text above cannot drift apart: they come from the same lines of code. Red, black, odd, even, high and low are not in the practice: a zero takes only half of those bets, and the level payout of a half is a different count.
What you will not see. No moment suggests a bet, and no moment tells you a cheaper bet is a good one. A shorter payout on level 3 is a thought experiment about arithmetic, not a claim about any table anywhere.
One strip, no rounds. Every square on the strip is one moment of this lesson, and all of the squares are there from the start. Your first answer fills the square; when the squares of one level are full, the next level begins by itself, with no button in between. Practising a mistake turns a cross into “fixed in practice” and does not change the count: only first answers do. Progress stays in this browser; nothing is sent anywhere.
Check yourself
Mayaformer floor supervisor · manners, money and breaks
Five quick ones. Pick an answer, then check it. No timer, no points: just whether it stuck.
The question we opened with. Neither: the single number and the dozen cost exactly the same. Both bets leave the same 2 pockets of the wheel unpaid, so both bets carry 5.26% of every unit bet. What differs between the two bets is only how the evening feels: the single number is paid rarely and pays a lot, the dozen is paid more often and pays a little. The price is identical, and that is the point: the price lives in the payout, not in the drama. Red would have been a different answer here: a zero takes only half of a bet on red, so red costs half as much.
Where to check this
18+ across Chile · Play responsibly · self-exclusion and the regulator are at the end of this page
Further reading
- Casino Math Guide: A Guide to Casino Mathematics · Robert C. Hannum, University of Denver (UNLV Gaming Studies Research Center)How the house advantage of a roulette wheel is worked out, wheel by wheel, and what it means as a price paid by the player.
- Teaching a university course on the mathematics of gambling · Stewart Ethier and Fred Hoppe, arXiv (2020)How a university course defines the house advantage: expected loss against the amount bet.
- Estándares para Máquinas de Azar · Superintendencia de Casinos de JuegoThe regulator’s technical standard: how the random number generator must behave, whatever the bet or the player.
- Funcionamiento de los juegos: Deberes y derechos de los jugadores · Superintendencia de Casinos de JuegoWhat a player may ask of the casino: the pay tables, a copy of the rules, a review of a recorded hand, and a complaint to the regulator.
Mayaformer floor supervisor · manners, money and breaks
One thing to carry out of here, and it is not about roulette. The price follows the number of bets you make, and the number of bets is the part of the evening you decide, before you sit down, while nothing is happening yet.
Knowing the price does not make an evening cheaper, and nothing here suggests that an evening at the wheel is a good idea for you. Knowing the price only means you are no longer paying it without knowing.
If it stops being fun
If gambling stops being fun, the SCJ self-exclusion applies in every casino in the country. Minimum entry age: 18 (age of majority under Law 19.221; Law 19.995 bars minors from gaming rooms). The casino may ask for an official ID. More on responsible gaming.
Show all regions
How it’s worked out
Sources, numbers and hosts
Numbers. Every figure on this page is worked out when the page is built, by our own code: a wheel of so many pockets, a bet covering so many of those pockets, a payout of so much. Nothing is typed in by hand, and each figure is recomputed by our checks from the engine rather than read back from this page.
Sources. The definition comes from arXiv (math.HO), preprint v2 (2020), and the two published prices, with the one bet that pays less than the rest, from UNLV Gaming Studies Research Center, quoted word for word behind the source buttons on this page, each with the date we read it. They are not decoration: the page refuses to build if the figure our own counting produces is not the figure in the quote. The layout of the wheel, and the rule that a zero takes half of an even-money bet, are quoted from the national catalogue of games published by Superintendencia de Casinos de Juego (SCJ), dated where they stand.
What is counted, and what is not. The payout a table gives a bet is part of the model this course computes with; the published price is what checks it. 35 to 1 is the only whole-number payout on a wheel of 38 pockets that produces the published 5.26%, and the same 35 to 1 is the only one on 37 pockets that produces 2.70%: one payout, two prices, because the counts differ. So the counting, the model and the published figure are three roads to one place.
The picture of a dozen. The twelve pockets marked for a dozen on this page are the first dozen, 1 to 12, because the price depends on how many pockets a bet covers and not on which ones.
The even-money bets. The half of the chip that comes back on a zero is the rule of the national catalogue, quoted behind the source button beside it; the price of 2.63% that follows from it is our count, because the catalogue prints no price.
What this page does not say. No bet is put forward here as a good one, and no system of any kind is either. A lower price is still a price, charged on every bet, and no arrangement of bets changes 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.
The price of a bet on a wheel with two zeros
Roulette Double-zero (except five-number) 5.26%UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read
What the house edge is
why the house advantage is defined as the ratio of expected loss to expected amount bet
The house advantage represents the long run percentage of the wagered money that will be retained by the casino.
Regardless of the method used to compute it, the house advantage represents the price to the player of playing the game.
UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read
The one bet on this wheel that costs more
Roulette Double-zero, five-number bet 7.89%UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read
One wheel, one price
Roulette Double-zero (except five-number) 5.26%UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read
The one bet on this wheel that costs more
Roulette Double-zero, five-number bet 7.89%UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read
A zero and the even-money bets
Si sale el cero (0) y el jugador ha apostado sobre una suerte sencilla pierde la mitad de su apuesta.Free translation: If the zero (0) comes up and the player has bet on an even-money bet, they lose half of their bet.
Si sale el doble cero (00) y el jugador ha apostado sobre una suerte sencilla pierde la mitad de su apuesta.Free translation: If the double zero (00) comes up and the player has bet on an even-money bet, they lose half of their bet.
Section I. Cuestiones generales; II.1 Ruleta francesa; II.2 Ruleta americana
What the house edge is
why the house advantage is defined as the ratio of expected loss to expected amount bet
The house advantage represents the long run percentage of the wagered money that will be retained by the casino.
Regardless of the method used to compute it, the house advantage represents the price to the player of playing the game.
UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read
One zero instead of two
Roulette Single-zero 2.70%UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read
The wheel, in the words of Superintendencia de Casinos de Juego (SCJ)
En el extremo de la mesa se encuentra el cilindro o rueda, en el cual se ubican 37 casillas, enumeradas del 1 al 36, además de un cero.Free translation: At the end of the table is the cylinder or wheel, which has 37 pockets, numbered from 1 to 36, plus a zero.
La Ruleta Americana con un cero significa que tiene, entre las suertes o posibilidades, el signo de un cero.Free translation: American Roulette with one zero means that, among its outcomes or possibilities, it has the sign of a zero.
La Ruleta Americana con doble cero significa que tiene, entre las suertes o posibilidades, el signo de un cero y además un doble cero.Free translation: American Roulette with double zero means that, among its outcomes or possibilities, it has the sign of one zero and also a double zero.
Section II.1 Ruleta francesa, 1.3.1; II.2 Ruleta americana, 2.3 Modalidades