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an explainer on the mathematics of chance

Games · Updated 17 August 2026

Roulette: a fixed edge built into the wheel

A wheel of 37 or 38 pockets paid as though it had 36. Every bet on the layout inherits the same shortfall.

A circle divided into thirty-seven equal pockets with one pocket shaded
The wheel has thirty-seven pockets. The payouts are calculated as though it had thirty-six.

One wheel, one number

The European wheel carries the numbers 1 to 36 and a single zero: 37 pockets. The American wheel adds a double zero: 38. The layout on the cloth, however, is priced on 36. A straight number pays 35 to 1, a split 17 to 1, a street 11 to 1, a corner 8 to 1, a column or dozen 2 to 1, and red, black, odd, even, high or low pay 1 to 1. Each of these payouts is exactly what would be fair if the extra pocket did not exist.

The consequence is unusually tidy: on a single-zero wheel every bet on the standard layout carries the same 2.70 per cent edge, and on a double-zero wheel almost every bet carries 5.26 per cent. Choosing a corner instead of a straight number changes how often a win arrives and how large it is, but not the fraction of stake retained.

Standard bets and their derived edge
BetNumbers coveredPayoutReturn on a 37-pocket wheelEdge
Straight135 to 136/37 = 0.97302.70%
Split217 to 12 x 18/37 = 0.97302.70%
Street311 to 13 x 12/37 = 0.97302.70%
Corner48 to 14 x 9/37 = 0.97302.70%
Dozen or column122 to 112 x 3/37 = 0.97302.70%
Red, black, odd, even181 to 118 x 2/37 = 0.97302.70%

The American exception

One bet on the double-zero layout is worse than the rest: the five-number line covering 0, 00, 1, 2 and 3, which pays 6 to 1. Its return is 5 x 7/38 = 0.9211, an edge of 7.89 per cent. It exists because five numbers do not divide neatly into a 36-based paytable, and the rounding was resolved in the house's favour.

Rules that halve the edge

Some European tables apply a rule on the even-money bets when zero appears: the stake is either returned in half or held for the next spin. Under the half-back version the even-money bets lose only half a unit rather than a whole one when zero lands, and the edge on those bets falls to 1.35 per cent. The rule changes nothing elsewhere on the layout; the straight number still returns 36/37.

Systems and physical wheels

Progressive staking plans are popular at roulette because the even-money bets win so often. They do not alter expectation. A plan that doubles after each loss wins a small amount very frequently and loses a very large amount rarely, and the two balance out to the same 2.70 per cent shortfall, with the added constraint that table limits and finite bankrolls cut the sequence short at the worst possible moment.

The one genuine exception is physical, not mathematical. A wheel with a manufacturing or wear defect can favour a section of the rim, which historically made careful observation of individual wheels worthwhile. Modern maintenance, wheel rotation and monitoring exist precisely because that vulnerability is real, and it is a property of a particular piece of equipment rather than of the game.

What this does not tell you

The figures assume a true wheel and the standard layout. Local variants, side bets and half-back rules change individual numbers, and no calculation here describes the result of one spin.

How these guides are worked out

Every figure on this site is derived from the stated rules of the game it describes, using elementary probability, and the arithmetic is printed beside the result so that a reader can check it. No gambling company, product or offer is named, rated or linked anywhere on the site, and nothing here is certified or assessed. These guides explain how the games work and why the arithmetic favours whoever sets the terms; they do not explain how to play, and they treat gambling as a subject rather than an activity to take up.

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