House edge, derived — not just asserted

An American wheel has 38 pockets: the numbers 1 to 36, plus 0 and 00. Bet one dollar on red: 18/38 ways to win, 20/38 ways to lose. The expected value works out to −2/38, which is −0.0526 — the textbook's own figure. In percentage terms that is a 5.26% house edge: for every hundred dollars cycled through the wheel, the long-run cost is a little over five dollars.

A European wheel has 37 pockets — a single zero. The same red bet now wins in 18 of the 37 pockets, and the straight-up number pays 35 to 1 against true odds of 1/37 (the payout table is the same one Massachusetts publishes in its official roulette rules: 35 to 1 on a single number, 1 to 1 on red). The expected loss per dollar comes to 0.027 — a 2.7% edge, roughly half the American wheel's. Same game, same felt, one pocket of difference; if you play roulette at all, the single-zero wheel is the only defensible choice.

The edge applies to every spin independently. Play long enough and the average grinds toward it — that is not bad luck, it is the definition. No staking system rearranges it: doubling after losses changes the shape of your ruin, not its expectation, as our martingale material shows in detail. The honest use of this number is comparative: between two games, the lower edge is the cheaper entertainment.