Roulette Odds and Payouts: European Wheel Explained

Roulette is a wheel with 37 pockets, numbered 0 to 36, and a ball that lands in one of them with equal probability. Every bet covers some of those pockets, wins if the ball lands in one, and pays in proportion to how few it covers. The house edge comes from the single green zero, and on a European wheel every bet carries exactly the same edge: 1/37, or 2.70%, at casino payouts. In Cashout Arcade the same wheel pays every bet at a 3% edge. This guide gives the exact chance and payout of every bet, and shows why no bet is a better value than any other.

The wheel and the layout

The wheel has 37 pockets: 18 red, 18 black and one green zero. The numbers 1 to 36 sit on a 3-by-12 table layout: the top row is 3, 6, 9 and so on, the middle row 2, 5, 8, and the bottom row 1, 4, 7. Each spin is independent, and the ball lands on each pocket with probability 1/37.

The table lets you cover pockets in different groupings:

The zero is the only pocket that beats almost every bet. When the ball lands on 0, a straight on 0 or a split involving 0 wins, and every other bet loses. That single losing pocket is the entire house advantage on a standard wheel.

Cashout Arcade offers these bets. It does not offer the 0-1-2-3 "first four" bet or the 0-1-2 and 0-2-3 trios, which a few casino tables have.

Exact chances and payouts

The chance of a bet is the number of pockets it covers divided by 37. Here is every bet with its exact chance, what a casino pays in the usual "to 1" terms, and what Cashout Arcade pays. (Our payouts include your stake, so "35.89x" means a 1-coin bet returns 35.89 coins, a profit of 34.89.)

Notice the structure: a casino pays 36 divided by the number of pockets covered, which is one coin less than the fair price of 37 divided by the pockets. In Cashout Arcade the rule is a single formula applied to every bet: the total return is 0.97 x 37 divided by the number of pockets covered, rounded down to four decimals. That formula is why the straight number pays 35.89 and not 36, and why our edge is a flat 3% rather than the 2.70% a casino wheel carries.

House edge: why every bet is the same

For a bet that covers k pockets at the casino return of 36 / k:

expected return = (k / 37) x (36 / k) = 36 / 37 = 97.30%

The k cancels. A straight bet, a split, a dozen and red-or-black all return exactly 97.297% on average, a house edge of 1/37 = 2.703%. At Cashout Arcade the return is (k / 37) x (0.97 x 37 / k) = 97%, again independent of k.

This is the key practical point about roulette. Unlike craps or baccarat, where the edge depends on which bet you place, roulette has no good or bad bet. The difference between a number and red-or-black is how often you win and how much you win, which is volatility, not value.

Single zero versus double zero

An American wheel adds a second green pocket (00) with the same payouts, so there are 38 pockets. The expected return of a straight bet becomes 36 / 38 = 94.74%, an edge of 2/38 = 5.263%, which is twice the European edge. According to Wikipedia's roulette article, the single-number expected value is -1 x 36/37 + 35 x 1/37 = -0.0270 on a European wheel and -1 x 37/38 + 35 x 1/38 = -0.0526 on an American one.

Cashout Arcade's wheel is single-zero. Many web pages misquote these figures (some give 2.63% for both), so the fractions above are worth keeping: 1/37 is 2.703%, not 2.63%.

Betting several things at once

Cashout Arcade's table lets you place several chips in one spin. The round's multiplier is the chip-weighted return: the sum of amount times pay for every bet that hits, divided by the total staked. Because every bet returns 97% on average, every layout does too, including covering every number.

An example. You place 100 on black, 50 on the second dozen (13 to 24) and 10 on number 17. That is 160 coins in total. The result depends on where the ball lands:

Across the 37 pockets, this layout loses everything on 13 numbers, returns 1.2461x on 12, 2.1808x on 5, 0.9346x on 6 and 4.4239x on one. Its average is 0.97, and it wins more than the stake on 18 numbers, which is 48.65%, the same as black alone. Mixing bets changes the shape of the payout, not the average.

Covering a lot of the table does lower the swings, which is useful if you want long sessions, but it also means more total coins wagered per spin, and the 3% edge applies to every coin of it.

Covering more numbers: chance up, payout down

A simple way to see the trade-off is to put equal chips on n different numbers. You win if the ball lands on any of them, and you are paid for the one that hits, minus the chips on the others:

Covering 36 numbers wins 97.3% of spins, but a win returns just under what you staked in total, because the one winning number pays 35.89 against 36 coins staked. Covering all 37 numbers guarantees a result of exactly 0.97x. That is the house edge with the dice removed: 3 coins in 100, certain. The more of the table you cover, the more the result looks like the plain cost of playing.

Why systems fail

Roulette attracts betting systems more than any other game: doubling after a loss, following streaks, betting on what is "due". They all fail for the same reason. The wheel has no memory, so each spin is a fresh 1-in-37 draw, and every bet on every spin has a negative expected value. Adding up negative-expectation bets, however cleverly timed or sized, gives a negative total. The betting systems guide shows the numbers, and the streaks guide shows how often long runs of one colour happen.

Responsible play

Roulette moves fast, and a table covered in chips can hide how much you have staked in total. Set a time limit before you start, decide the coins you are willing to spend as entertainment, and stop when it stops being fun. The wheel does not remember, and a loss now does not make a win more likely later.

You can play European single-zero roulette in the Roulette game, and every game's return is on the Odds page.