The wheel layouts
A roulette wheel has numbered pockets, and a ball lands in exactly one of them. The layout decides every probability that follows.
- European wheel: 37 pockets, numbered 1 to 36 plus a single zero.
- American wheel: 38 pockets, numbered 1 to 36 plus a zero and a double zero.
The numbers 1 to 36 are split evenly between red and black (18 each), odd and even (18 each), and low (1-18) and high (19-36). The zero pockets are green and belong to none of those groups. That uncounted space is what produces the margin.
Probability of each bet type
Assuming a fair wheel, each pocket is equally likely, which is a classic case of equally likely outcomes. A bet’s probability is the number of pockets it covers divided by 37 (European) or 38 (American).
| Bet | Pockets covered | European probability | Standard payout |
|---|---|---|---|
| Straight up | 1 | 1/37 (2.70%) | 35 to 1 |
| Split | 2 | 2/37 (5.41%) | 17 to 1 |
| Street | 3 | 3/37 (8.11%) | 11 to 1 |
| Corner | 4 | 4/37 (10.81%) | 8 to 1 |
| Six line | 6 | 6/37 (16.22%) | 5 to 1 |
| Dozen or column | 12 | 12/37 (32.43%) | 2 to 1 |
| Red, black, odd, even, high, low | 18 | 18/37 (48.65%) | 1 to 1 |
Each payout follows a simple pattern. The fair price of covering n pockets out of 36 non-zero numbers is (36 - n)/n to 1, and the game pays exactly that, as if the zero did not exist. A straight-up bet fair-pays 35 to 1 against 36 numbers, a split pays 17 to 1 (because 34/2 = 17), and so on.
The margin, bet by bet
Expected value per unit staked is: (probability of winning x payout) - (probability of losing x 1).
On a European wheel:
- Straight up: (1/37 x 35) - (36/37) = (35 - 36)/37 = -1/37
- Corner: (4/37 x 8) - (33/37) = (32 - 33)/37 = -1/37
- Dozen: (12/37 x 2) - (25/37) = (24 - 25)/37 = -1/37
- Red: (18/37 x 1) - (19/37) = -1/37
Every one comes to -1/37, which is about -2.70%. The margin is the same no matter which bet is placed. Spreading chips across the layout, or picking “safer” bets, does not reduce the average loss; it only changes how that average is reached.
On an American wheel the winning probabilities fall to n/38 while payouts stay the same. A straight-up bet becomes (1/38 x 35) - (37/38) = (35 - 37)/38 = -2/38, about -5.26%. The same percentage applies to the others, with one well-known exception: the five-number bet (0, 00, 1, 2, 3) found on some American layouts carries a larger margin of about 7.89% (-3/38).
Why the American wheel is roughly twice as costly
Adding one more pocket that no ordinary bet covers doubles the uncovered space from 1/37 to 2/38. The wheel has the same appearance and the same payout list, but a player’s average loss per unit staked is about twice as high. Anyone comparing tables should look first at how many zeros are on the wheel.
Rule variants that change the edge
Some European-style tables use rules that soften the zero’s effect on even-money bets:
- La partage: if the ball lands on zero, an even-money bet loses only half the stake. The expected value for those bets becomes: (18/37 x 1) - (18/37 x 1) - (1/37 x 0.5) = -0.5/37, about -1.35%.
- En prison: the even-money stake is held for the next spin. The effect on the average is similar to la partage.
These rules apply only to even-money bets. Inside bets such as straight up and corners keep their normal margin.
Streaks, patterns and the wheel’s lack of memory
Each spin is one of a series of independent events, so the ball does not remember earlier results. A run of seven reds does not make black more likely on the next spin, and displayed result boards do not reveal useful patterns on a fair wheel. The belief that they do is the gamblers fallacy, and it is one of the most common errors at roulette tables. Equally, runs are normal: with a 48.65% chance of red on each spin, long runs of one colour appear regularly in a large number of spins.
A wheel with a physical fault could in principle produce uneven results, which is something a chi-square test on a large record of spins can examine, but regulated venues typically maintain their equipment, and any such check would need a very large sample to detect small bias.
What changes between bets: variance, not margin
Because every bet has the same expected loss per unit, the real difference between bets is how the results are spread out. A straight-up bet wins rarely and pays a lot; a red bet wins nearly half the time and pays little. Over 100 one-unit bets on a European wheel, the expected loss is 2.70 units for both. The standard deviation is about 10 units for red and about 58 for straight up. That difference in spread is the idea of game volatility rather than of margin.
Roulette outcomes are uncertain, and the figures here describe averages over many spins, not what will happen in any one visit.