In brief
Roulette is a table game in which the croupier launches a ball onto a spinning wheel and the players bet on the pocket where the ball will come to rest. Every pocket has the same chance, and the payouts are fixed in advance. The mathematics of the game comes down to a single observation: the payouts are calculated as if the zero did not exist. That is where the house edge comes from, and it is the same for almost every bet.
The wheel and the table
The European wheel has 37 pockets: the numbers 1 to 36, half red and half black, plus the zero, coloured green. The numbers are not in order around the wheel: red alternates with black, and low and high, odd and even numbers are spread as evenly as possible.
The croupier spins the wheel in one direction and launches the ball the other way, along its rim. Bets can usually still be placed while the ball is travelling, until the croupier announces that betting is closed ("rien ne va plus"). The winning number is the pocket where the ball stays. The croupier marks it on the table, collects the losing stakes and pays the winning bets.
The betting layout repeats the numbers 1 to 36 in a grid of 12 rows and three columns, with the zero at one end. Around the grid are the areas for bets on large groups: the dozens, the columns, red and black, odd and even, 1–18 and 19–36. Many tables also have an oval track that reproduces the order of the numbers on the wheel, used for bets on sections of the wheel.
Three variants
European roulette
One zero and 37 pockets. It is the usual variant in Europe and the one most figures on this page refer to.
American roulette
The wheel has 38 pockets: besides the zero there is a second green pocket, the double zero (00). The payouts are the same as in European roulette, so the second green pocket brings the player nothing; it only increases the house edge. The American layout also has a bet on five numbers (0, 00, 1, 2 and 3), paid 6 to 1.
French rules
French roulette uses the single-zero wheel and adds a rule for the even-money bets (red or black, odd or even, 1–18 or 19–36). Under "la partage", when the zero comes up, the player loses only half of a stake on an even-money bet and the other half is returned. A related rule, "en prison", keeps the stake "imprisoned" for the next spin: if the bet wins then, the player gets the stake back, without winnings. Not every single-zero table applies these rules, and the details differ from one venue to another. The French names of the bets (plein, cheval, transversale, carré, sixain) also come from the French table.
The bets
"Inside" bets are placed on numbers in the grid, "outside" bets on the areas around it. The table below shows the basic bets in European roulette: how many numbers each covers, what it pays, its chance of winning on one spin and its average result per 100 units staked.
| Bet | Numbers covered | Pays | Chance of winning | Average result per 100 units |
|---|---|---|---|---|
| Straight up | one number | 35 to 1 | 2.70% | −2.70 |
| Split | two numbers | 17 to 1 | 5.41% | −2.70 |
| Street | three numbers | 11 to 1 | 8.11% | −2.70 |
| Corner | four numbers | 8 to 1 | 10.81% | −2.70 |
| Six line | six numbers | 5 to 1 | 16.22% | −2.70 |
| Dozen or column | 12 numbers | 2 to 1 | 32.43% | −2.70 |
| Even-money bet | red/black, odd/even, 1–18/19–36 | 1 to 1 | 48.65% | −2.70 |
- Straight up (plein): a single number, zero included.
- Split (cheval): two numbers next to each other on the layout.
- Street (transversale): a row of three numbers.
- Corner (carré): four numbers that meet at a corner.
- Six line (sixain): two adjacent rows, six numbers.
- Dozen or column: 1–12, 13–24 or 25–36, or one of the three columns of the grid.
- Even-money bets: red or black, odd or even, 1–18 ("manque") or 19–36 ("passe"). The zero belongs to none of them, so, outside the French rules, all these bets lose when the zero comes up.
A payout of 35 to 1 means the player receives 35 units for each unit staked and keeps the stake. The announced bets on the oval track, such as "neighbours of zero" or "orphans", are simply packages of basic bets placed together on a section of the wheel; their mathematics is that of the bets they are made of.
The same bets on the American wheel:
| Bet | Numbers covered | Pays | Chance of winning | Average result per 100 units |
|---|---|---|---|---|
| Straight up | one number | 35 to 1 | 2.63% | −5.26 |
| Split | two numbers | 17 to 1 | 5.26% | −5.26 |
| Street | three numbers | 11 to 1 | 7.89% | −5.26 |
| Corner | four numbers | 8 to 1 | 10.53% | −5.26 |
| Six line | six numbers | 5 to 1 | 15.79% | −5.26 |
| Dozen or column | 12 numbers | 2 to 1 | 31.58% | −5.26 |
| Even-money bet | red/black, odd/even, 1–18/19–36 | 1 to 1 | 47.37% | −5.26 |
The five-number bet is missing from the table because it is the exception on the American layout: its house edge is 7.89%, higher than any other bet on the same table.
Where the house edge comes from
Roulette payouts are the fair ones for a wheel that would have only the numbers 1 to 36. On such a wheel, a straight-up bet paid 35 to 1 would be fair: over 36 spins, a player putting one unit on the same number each time would, on average, win once (35 units) and lose one unit 35 times. The zero adds a 37th pocket, and the payout stays the same. Over 37 spins the player still wins once on average, but loses 36 times: one unit down for every 37 units staked. That is the house edge in European roulette: 1/37 of every stake, or 2.70%.
The same happens with every bet. A dozen pays 2 to 1: on a wheel without a zero, 12 winning numbers against the other 24 would make 2 to 1 exactly the fair price. The zero joins the losing numbers, and the payout ignores it. However many numbers a bet covers, the zero adds the same average loss, 1/37 of the stake. This is why, on the European wheel, the last column of the table has the same value on every row: no bet is "better" than another in terms of average result. Only how often it wins and how much it pays differ.
On the American wheel, the two green pockets double the effect: the house edge is 2/38, or 5.26%, on every bet in the American table. The la partage rule halves the loss on the zero for even-money bets, so their house edge falls to 1.35%; on the other bets it stays at 2.70%. The rule reduces the edge but does not remove it: in every variant, the edge stays with the house.
Seen from the player's side, the same figure is expressed as the return to player (RTP): 97.30% in European roulette and 94.74% in American roulette. The link between the two ideas is explained in the article on the house edge and RTP.
An example: 100 units staked
Over an evening, a player places bets that add up to 100 units (100 lei, for instance) on a European wheel. How they are spread does not matter: numbers, colours or dozens. The player's average result is −2.70 units.
"Average" means the average over a very large number of such evenings. In a single evening the result can be very different: the player may finish ahead or lose everything. What differs from one bet to another is not the average but the spread of results. The standard deviation of a straight-up bet is 5.84 units for every unit staked, against 1.00 for an even-money bet. The larger the deviation, the longer the losing runs and the larger the occasional wins; the details are in the article on variance and the law of large numbers.
The amount in the example is the total of the stakes, not the money brought to the table. A player who puts winnings back on the table increases the total staked, and the average loss grows in the same proportion. It depends on how much is staked in total, not on how much the player has in their pocket.
Spins are independent
The wheel has no memory. After nine reds in a row, the chance that the tenth spin is red is still 48.65%, exactly as on any other spin. The probability that ten particular spins, chosen in advance, all come up red is small, 0.0743%, but a wheel spins a great many times, so such runs do appear now and then. Once the first nine have happened, they change nothing for the tenth.
The boards that display the latest numbers show the past, not the future. The idea that a colour or a number "is due" after a long absence is a well-known error of reasoning, described in the article on the gambler's fallacy. For the same reason, no system of doubling or splitting stakes changes the house edge: each stake, taken on its own, has the same negative average result, as the article on betting systems shows.
In electronic roulette the winning number comes from a random number generator instead of a ball. If the wheel has the same pockets and the payouts are the same, the mathematics is identical.
Terms
- House edge: the share of each stake the house keeps on average.
- RTP: the share of stakes that returns to players on average.
- Variance: how widely results are spread around the average.