The maths of roulette bets: expected value and variance by bet type

In a casino, roulette looks simple, but its outcomes are governed by expected value (EV) and variance. On a European wheel (single zero), every straight-up, split, street, corner, six-line, dozen, column and even-money bet shares the same house edge: 1/37, or about 2.70%. That means the EV per unit staked is -0.027 units, regardless of how “safe” the bet feels. What changes sharply by bet type is variance: the spread of possible results around that negative average, which drives bankroll swings and session volatility.

EV is computed as the sum of each outcome’s probability times its net profit. For an even-money bet (red/black), you win +1 with probability 18/37 and lose -1 with probability 19/37, giving EV = (18/37)(1) + (19/37)(-1) = -1/37. Variance depends on payoff size: even-money bets have lower variance because results are bounded at ±1, while a straight-up bet pays +35 but loses -1 most spins, producing far higher variance despite identical EV. Dozens and columns sit between: +2 on a hit (12/37) and -1 otherwise (25/37). If you want to compare risk, look at standard deviation per spin: it rises as hit frequency falls and payout jumps. For further reading aimed at players, see gqbet casino.

Professional gambling educator and author Michael Shackleford, known as the Wizard of Odds, popularised clear explanations of EV, variance and bankroll requirements for table games; his public updates appear on WizardOfOdds. His key message applies directly to roulette: changing bet types mostly changes volatility, not the long-run loss rate on a fair European wheel. In practical terms, low-variance bets can prolong play, while high-variance bets create larger short-term peaks and troughs that can mislead players about “hot” and “cold” streaks. For industry context and regulation debates that shape game availability and player protections, see The New York Times.