Wheel games

A roulette wheel is the most transparent machine on any gaming floor. It has a fixed number of pockets, every pocket is intended to be equally likely, and the table pays fixed odds on bets covering one or more of them. There is no strategy, no skill and no information: every spin is the same experiment repeated, which makes it the ideal object for showing what a house edge is.

The single-zero layout has thirty-seven pockets, numbered one to thirty-six plus a zero. A straight-up bet on one number wins once in thirty-seven spins and pays thirty-five to one. The honest price would be thirty-six to one. One unit of price is withheld, which over thirty-seven spins is one unit in thirty-seven: 2.70 per cent.

What is elegant, and slightly unsettling, is that this figure does not change if you spread the bet out. An even-money bet covers eighteen pockets of thirty-seven and pays one to one; a split covers two and pays seventeen to one; a dozen covers twelve and pays two to one. Work any of them out and the answer is 2.70 per cent. The layout was constructed so that every bet is priced as if the zero did not exist, which means the zero costs the same wherever you stand.

The double-zero layout adds a second zero without changing any payout. The straight-up bet now wins once in thirty-eight and still pays thirty-five, so two units of honest price are withheld instead of one: 2/38, or 5.26 per cent. The game looks identical and costs almost twice as much, which is the clearest demonstration available that a game's cost lives in its pay table rather than its appearance.

The one bet that breaks the pattern exists only on the double-zero layout: the five-number bet covering zero, double zero, one, two and three, which pays six to one. Five pockets in thirty-eight win, thirty-three lose, so the expected value is (5 × 6 − 33)/38 = −3/38, or 7.89 per cent. It is the worst bet on the wheel and the only one that is worse than the rest, purely because five does not divide neatly into thirty-six.