Wheel and dice
The wheel: two layouts, two numbers, one method
A wheel is the cleanest game to analyse, because its sample space is printed on it. Count the pockets, read the payout, subtract.
| Layout | Numbered pockets | Zero pockets | Total pockets | Single-number payout |
|---|---|---|---|---|
| Single zero | 36 | 1 | 37 | 35 to 1 |
| Double zero | 36 | 2 | 38 | 35 to 1 |
The wheel is the game where the mathematics is least able to hide. Every outcome is equally likely, the outcomes are visible, and the payout schedule is fixed and public. There is no strategy layer to complicate the count, which is why it is the natural starting point for anyone working out what an edge is.
The standard European layout has thirty-seven pockets: the numbers one to thirty-six and a single zero. The standard American layout adds a second zero, making thirty-eight. In both layouts, a straight bet on a single number pays thirty-five to one. The payout does not change when the pocket count does, and that is the entire difference between the two games.
Working, single-zero layout
37 pockets, 1 winning, 36 losingexpected value = (1/37 x 35) + (36/37 x -1)= (35 - 36) / 37 = -1/37 = -0.027027edge = 2.70%Working, double-zero layout
38 pockets, 1 winning, 37 losingexpected value = (1/38 x 35) + (37/38 x -1)= (35 - 37) / 38 = -2/38 = -0.052632edge = 5.26%Why nearly every bet on a single-zero wheel gives the same answer
A player who dislikes the straight bet can spread across two numbers, a column, a dozen or an even-money proposition. The payouts change accordingly: seventeen to one on a split, two to one on a column, one to one on red or black. It makes no difference at all to the percentage, because the payouts were constructed by taking the thirty-six numbered pockets as if they were the whole wheel and then paying accordingly, while the zero sits outside every one of those propositions.
Working, three different bets on the same 37-pocket wheel
split, 2 numbers, pays 17 to 1: (2/37 x 17) + (35/37 x -1) = (34 - 35)/37 = -1/37column, 12 numbers, pays 2 to 1: (12/37 x 2) + (25/37 x -1) = (24 - 25)/37 = -1/37even-money, 18 numbers, pays 1 to 1: (18/37 x 1) + (19/37 x -1) = (18 - 19)/37 = -1/37all three = 2.70%That uniformity is the design working as intended. It means no arrangement of chips on the cloth improves the expectation, and it means the wheel's edge can be quoted as a single number without qualification. It also disposes of the whole family of systems built on covering more of the layout. Covering more numbers reduces the size of the swings, which some players genuinely prefer, but it moves the expectation not at all.
The one bet that is worse
The double-zero layout carries a fifth proposition that has no equivalent on the single-zero wheel: a line covering five pockets, paid at six to one. Because five does not divide the thirty-six-pocket payout scheme cleanly, the rounding falls against the player harder than anywhere else on the table.
Working, five-number line on a 38-pocket wheel
5 winning pockets, 33 losing, pays 6 to 1expected value = (5/38 x 6) + (33/38 x -1)= (30 - 33) / 38 = -3/38 = -0.078947edge = 7.89%What the wheel does not remember
Wheel spins are independent trials. A pocket that has appeared five times in succession is exactly as likely on the next spin as any other, because nothing about the wheel's physical state carries information about its history. The recorded sequences displayed beside many tables are accurate records of the past and contain no information about the future. They persist because people find patterns compelling, not because the patterns predict.
The single caveat is physical rather than mathematical. A wheel that is worn, tilted or unbalanced is no longer a uniform sample space, and historically some wheels were not uniform. That is a manufacturing and maintenance question, and it is one of the reasons that equipment testing became a regulatory concern. It is not a strategy, and a correctly maintained modern wheel is designed specifically to be indistinguishable from the uniform model above.