Expected Value in Roulette

Roulette is a simple game with a mathematical edge that favors the house.

DefinitionAmerican Roulette

The wheel has 38 pockets: numbers 1-36, 0, and 00. A bet on a single number pays 35:1.

TheoremExpected Value

The expected value of a $1 bet on a single number is:

E = \frac{1}{38}(35) + \frac{37}{38}(-1) = -\frac{2}{38} \approx -\$0.0526

DefinitionEuropean Roulette

The wheel has 37 pockets: numbers 1-36 and 0 (no 00).

TheoremEuropean Expected Value

E = \frac{1}{37}(35) + \frac{36}{37}(-1) = -\frac{1}{37} \approx -\$0.027

Remark

The European wheel has roughly half the house edge of American roulette. Always prefer European when available.