Atlassino Academy · How casinos work · Lesson 5
One night versus a thousand
An hour of bets on a dozen ends in front about four times in ten. A hundred hours almost never does. Same wheel, same bet — the only difference is how many times the house edge has been charged.
Lesson 5 of the how-casinos-work track. You already know the house edge (the share of all the money bet that the casino keeps on average, over many rounds). Charged on every spin of an hour, it adds up to the price of an hour (bet, times rounds an hour, times the house edge: what an hour of a game costs on average). Now the awkward part: the house edge is taken out of every single spin, and people still walk out in front all the time — not once in a blue moon, but on an ordinary Friday. So what does an evening that ended in front actually tell you about the person who played it?
You and the person beside you play the same hour at the same wheel, the same bet. You end behind; they end in front. Which of you played the hour better?
Your hour8 bets behind
The hour beside you7 bets in front
Two real hours at the same double-zero wheel, 38 spins each, the same bet on a dozen. Only the order of the pockets differed: one tick a spin, coloured in for a spin that paid.
The short answer
An hour of bets on a dozen at a double-zero wheel ends in front 42.3% of the time. Stretch that same flat bet to a thousand spins and the share still in front falls to 11.4%. Stretch the same bet to a hundred hours and the share still in front is 1.0%. Not one of those three shares has anything to do with the person holding the chips.
Leoformer dealer · rules and maths
I dealt for years, and the sentence I heard most at the end of a shift was some version of “tonight worked”. It never did work, and it never could: one night is a single draw out of a very wide bag, and it is the same bag whoever reaches into it.
That is why I would rather look at the same hour a hundred times over than at any single hour, including the one I have just dealt you. Ten evenings of one person tells you a little more than one evening does. A few thousand bets in a row tells you nearly everything — and what it tells you is about the wheel, not about the person making the bets.
One hour, a hundred times
Start with a bet the zeros take whole: a dozen. Put a chip on the first dozen: if the ball lands on a number from 1 to 12, the wheel hands your stake back and twice as much again, and if the ball lands anywhere else, the zero and the double zero included, the chip is gone. Red or black works differently here: in the national catalogue a zero takes only half of a bet on red or black, so red or black costs half as much as a dozen, and this page stays with the dozen. Here are all 38 pockets the ball can land in, one block each.
12 pay you24 are the other two dozens and take the chip2 green, and green takes it too
That green pair, and nothing else in the whole machine, is the 5.26% the house keeps on this bet.
A wheel also has a pace, and the pace has a name: the rounds an hour (how many rounds a table gets through in an hour, which is what turns a bet into an hourly cost). At the example pace this course uses, a double-zero wheel gets through 38 spins an hour, and the house edge is charged on each of those spins separately — not once at the start of the evening, and not once at the end. Play that same hour a hundred times over, and the hundred hours land like this.
in front 42.3%behind 57.7%
A hundred hours at the same wheel, the same flat bet on every spin. The average result of an hour at this wheel is 2 bets behind, and finishing 2 bets behind is also the single commonest way for one of these hours to end: 13.8% of the hundred hours land on that exact result.
Leoformer dealer · rules and maths
Look at that bar again, because the coloured part of it — the hours that ended in front — is the part that gets narrower every time the run of the same flat bet gets longer. About four hours in ten ending in front is not a loophole somebody clever found. It is what a bag with two green pockets in it does to a hundred people who all did exactly the same thing.
Which result is the ordinary one
Go back to those two hours at the wheel: one ended 7 bets in front, the other 8 bets behind. On a dozen an hour of 38 spins cannot end the same distance either side of level, so this is as close to a mirror pair as an hour gets. Are the two hours mirror images of each other?
7 bets or more behind30.6%
7 bets or more in front19.0%
How often an hour at that wheel ends at least that far from level, counted behind and counted in front. The two bars are drawn on the same scale, so the longer bar is the commoner hour.
An hour that far from level is about one and a half times likelier to be behind than in front, and that lean is the house edge: it tilts the whole picture of a hundred hours a little to the left.
| At least this far from level | Hours behind | Hours in front |
|---|---|---|
| 4 bets | 43.9% | 29.5% |
| 10 bets | 19.3% | 11.2% |
| 13 bets | 10.8% | 6.1% |
Every row is the same hour at the same wheel; only the distance from level changes from row to row. Read the two columns down together and both columns thin out, but the hours in front thin out faster than the hours behind: the further from level you look, the more the hours behind outnumber the hours in front.
So an hour that ended in front gets retold for weeks, and an hour that ended behind is forgotten by Tuesday. Ending in front is simply the rarer of the two endings — and being rare is not the same thing as being earned, because nothing either person at that wheel did changed which pockets the ball found.
What a thousand bets does to the picture
Two things grow as an evening gets longer, and they do not grow at the same speed. The first is the loss, and the loss keeps exact step with the spins: every spin is charged the same house edge, so the average result slides further behind level in a straight line.
The second thing that grows is the swing: how far an ordinary run happens to wander either side of the average result. The swing grows much more slowly than the loss. Early on the swing is wide enough to cover the loss completely; later it is not, and here is what that looks like at five lengths of run.
38 spins · one hour42.3% in front
380 spins · 10 hours23.6% in front
1,000 spins · 26 hours11.4% in front
3,800 spins · 100 hours1.0% in front
10,000 spins · 263 hours0.0% in front
in frontbehind
The coloured part of each bar is the runs still in front. A thousand spins is about 26 hours at a table, which is an hour every Friday night for half a year.
None of this means that a losing run is due a good one: a wheel has no memory at all — Do cards, dice and wheels remember? is the one next door. The house edge never changed; what ran out is the room the swing had to hide the house edge in.
The cheaper the bet, the longer it hides
A much cheaper even-money bet sits a few steps away: the pass line at craps, the bet you put down before the first roll of a round. It charges 1.41% of each bet, where a dozen at a double-zero wheel charges 5.26%.
A double-zero wheel, a dozen5.26% of each bet
The pass line at craps1.41% of each bet
Drawn on the same scale: the pass line costs roughly a quarter of what a dozen at a double-zero wheel costs.
Lining the two bets up against each other takes one more step, because a spin of a wheel settles in that one spin while a single pass-line bet can take several rolls of the dice to settle. An hour of clock time is therefore not the same amount of play at the two tables.
One hour at a double-zero wheel38 spins
One hour at the craps pass line18 settled bets
An hour at the craps table settles fewer bets than an hour at the wheel spins. So the two prices can only be set against each other bet for bet, and never hour for hour.
| Bets | In front: a dozen, 5.26% | In front: pass line, 1.41% |
|---|---|---|
| 100 | 33.6% | 40.5% |
| 380 | 23.6% | 37.2% |
| 1,000 | 11.4% | 31.6% |
| 3,800 | 1.0% | 18.7% |
| 10,000 | 0.0% | 7.7% |
At the dozen 1.0% of runs are still in front after 3,800 bets; at the pass line 18.7% still are.
The pass line and the dozen end in the same place, only later: a cheaper price buys a longer disguise, and a long, handsome record at a cheap bet is the easiest thing in the room to mistake for skill. Neither bet is a recommendation here.
Practice: a hundred evenings
Each round of practice shows finished play in full, either one person’s or three people’s at once: the individual spins as a row of ticks above, where there is room to draw them, and underneath a running count of how far in front or behind the play stood after every spin. Your job is to read that finished run against the hundred runs of the same length it was drawn from. Level 1 shows three runs of three different lengths, all three ending in front; level 2 shows ten evenings of one person; level 3 shows the same reading at a bet that charges a quarter as much.
Only level 1 is scored, and every square on the strip is one run of level 1. All the squares are there from the start, so the strip never grows or shrinks under you. When every square has been filled in, level 2 begins by itself. Levels 2 and 3 have no squares at all, because there the question has the same answer every time, and answering the same question ten times over would measure patience rather than understanding.
Lesson done
Review my mistakes (0)
Progress stays in this browser; nothing is sent anywhere.
The practice table needs JavaScript. Here is one run from it, with the answer.
Level 1 · Three runs
A double-zero wheel · the same flat bet on a dozen · three runs, all three ending in front
Same wheel, same flat bet, three different lengths of run.
76 spins (2 hours)5 bets in front
38 spins (one hour)1 bet in front
190 spins (5 hours)11 bets in front
Three people played this wheel with the same flat bet, and all three finished in front. Which of the three endings in front is the most surprising one?
The answer: 190 spins (5 hours)
The longer the run, the less an ending in front says about the person who played it. Out of a hundred runs: 42.3% at 38 spins, 35.1% at 76 spins, 29.0% at 190 spins. Same wheel, same flat bet, the same house edge charged on every single spin — the only thing that differs between the three runs is how many times that house edge has been charged.
in front 29.0%behind 71.0%
Level 2 · Ten Fridays
The same wheel and the same bet on a dozen, ten Fridays, one hour each · 38 spins an hour
Ten Fridays at the same wheel, an hour each.
On the total52 bets in front
After ten Fridays he is in front on the total, and he wants to make Friday at this wheel a habit because of that total. What does a ten-Friday record tell you about the way he plays?
The answer: Nothing about the way he plays.
Ten hours is 380 bets, and 23.6% of runs that long end in front of where they started — about 24 people in every hundred who play ten hours. Ten more evenings of the same length cost about 200 units on average at an example bet of 10 units, which is the average cost of one hour — the bet, times the spins in an hour, times the house edge — ten times over.
in front 23.6%behind 76.4%
Level 3 · A cheaper bet
The pass line at craps · 18 decisions · one hour
An hour at the pass line, and the hour ended in front.
One hour2 bets in front
The pass line charges about a quarter of what a dozen at a double-zero wheel charges. This hour at the pass line ended in front. Does an hour in front here say more about the person who played it than the same hour at the wheel would, or less?
The answer: Less than the same hour at the wheel.
At the pass line the house charges 1.41% of each bet against 5.26% on a dozen at the wheel, and the difference shows in how long the swing keeps the loss out of sight. After 1,000 bets, 31.6% of runs at the pass line are still in front against 11.4% of runs at the wheel; after 10,000 bets the two shares are 7.7% at the pass line against 0.0% at the wheel. Both bets end in the same place. The pass line takes far longer to get there, and that is what makes a long, good-looking record at a cheap bet the easiest thing in the room to mistake for skill.
in front 38.4%exactly level 18.5%behind 43.1%
Every run in the practice table is worked out the same way: the run itself is a real draw from the outcome table of that bet, and the hundred runs it is read against are the exact distribution of that many bets rather than a hundred more draws.
How the practice table works
Where the runs come from. Every run is a real draw from the outcome table of that bet, drawn when the page is built, so the picture and the count cannot disagree. Runs are drawn until one ends the way the moment needs (all three in front on level 1, in front on the total on level 2), because that is the case being read; the order of spins inside a run is never arranged. Past sixty spins only the line is drawn — sixty ticks is about as many as a phone can show and still be read.
The hundred runs beside it are not a hundred more draws but the exact distribution of that many bets. A hundred draws would land near the true figure rather than on it, and the explanation would then be naming a number the picture did not show.
What is scored. Level 1, and nothing else. Levels 2 and 3 are a reference you page through: neither teaches a way of playing that a progress bar could measure.
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.
And the question at the start
Neither of you played the hour better, and the question has no other honest answer. Out of a hundred hours played exactly that way, 42 finish in front and the rest finish behind. You and the person beside you drew two hours out of that hundred. The only thing that differed between the two of you was the order in which the pockets came up, and neither of you had any say in that order.
Where to check this
18+ across Chile · Play responsibly · self-exclusion and the regulator are at the end of this page
Further reading
The documents this track is written from, in full.
- 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
Here is the practical half of this lesson. An hour that ended in front and an hour that ended behind, at the same wheel and at the same flat bet, are two draws out of the same hundred hours, and the order the pockets came up in is the whole of the difference between them. So write down the hours and the money you meant to spend before you leave the house, and the next day read what you wrote instead of reading the result. The result was never going to tell you anything about the person who played it.
And what one of those hours costs in money rather than in bets is the bet, times the spins in an hour, times the house edge: about 20 units an hour at 10 units a spin, on average.
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
Sources. All of them sit behind the ⓘ buttons in the text: the definition of the house edge, from arXiv (math.HO), preprint v2 (2020); the published edge of a double-zero wheel, from UNLV Gaming Studies Research Center, which is the price on a wheel where a zero takes the whole bet; and the catalogue of games Superintendencia de Casinos de Juego (SCJ) publishes, where a zero takes the whole of a bet on a dozen and only half of a bet on red or black. That is why every figure on this page is for a dozen: on red or black in Chile the published figure would not hold. No casino or regulator we could read publishes how many spins an hour a table gets through, so the 38 spins an hour on this page are an example pace, not a measured one, and the page says so rather than implying a local figure.
Numbers. Every figure is worked out when the page is built. The outcome tables of the two bets are counted from the pockets of the wheel and the faces of the dice, by the same code as our other figures: a dozen pays twice the chip on 12 pockets of 38 and loses it on the rest. The share still in front after a given number of bets is the exact distribution of that sum, not a simulation of it. The shares count bets, never money: the only money on this page is the average cost of an hour, which is the bet, times the spins in an hour, times the house edge.
The two example hours at the start. Both are real draws from the same outcome table as everything else here, drawn when the page was built, and then picked for their endings: one hour that finished exactly 8 bets behind and one that finished exactly 7 bets in front, the closest an hour on a dozen comes to the same distance either side of level, because that pair of endings is the thing the lesson compares. The order of spins inside each of the two hours is exactly as it was drawn, and no spin in either hour has been moved.
The swing. How wide the swing is has no figure anywhere on this page, in bets or in money. Putting a number on it needs a measure of spread, and that measure, along with what an evening can do to a set amount of money, belongs to the money-and-risk track. Here the swing is named, and what it does is shown by the share of runs still in front.
The model, stated plainly. Every figure here assumes the same flat bet on every round, exactly that many rounds, no change of size and nobody stopping early. Real evenings do none of those things, so these are shares of runs of this model — not shares of people. We have no figure for what share of visitors leave in front, and no operator we could read publishes one.
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. Nothing on this page says that a cheaper bet is a good idea, or that a longer stretch of play is, or that an evening at a casino is. Every column in every table here ends in the same place.
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
What a zero takes
Si sale el cero (0) y el jugador ha apostado a otras apuestas que no cubren el cero (0), pierde su apuesta.Free translation: If the zero (0) comes up and the player has placed other bets that do not cover the zero (0), they lose their bet.
Si sale el doble cero (00) y el jugador ha apostado a otras apuestas que no cubre el doble cero (00), pierde su apuesta.Free translation: If the double zero (00) comes up and the player has placed other bets that do not cover the double zero (00), they lose their bet.
Section I. Cuestiones generales; II.1 Ruleta francesa; II.2 Ruleta americana
A zero takes half of an even-money bet
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
The house edge at a double-zero wheel
Roulette Double-zero (except five-number) 5.26%UNLV Gaming Studies Research Center · Hannum, Casino Math Guide: A Guide to Casino Mathematics · read