Roulette mathematics: expected value, house edge and volatility

Core Roulette concepts: house edge, expected value and standard deviation

The most important parameters to consider when analyzing a game are:
HA (House Advantage). For a bet with only two possible outcomes — a net win paid at x to 1 or the complete loss of the stake — if y to 1 represents the fair odds against winning, the house edge is calculated as:
HA = (y − x) / (y + 1)
To express it as a percentage, the result must be multiplied by 100. The formula does not apply directly to bets with an intermediate outcome, such as the return of half the stake under La Partage.

EV (Expected Value). Expected value measures the theoretical average result per unit wagered and includes every possible outcome:
EV = Σ (xᵢ × pᵢ)
where xᵢ is the net result of outcome i and pᵢ is its probability. An EV of −0.027 means, for example, a theoretical average loss of 0.027 units for every unit wagered in total.

Standard deviation. Standard deviation measures the dispersion of results around their expected value:
σ = √[Σ pᵢ(xᵢ − EV)²]
If the bets are independent and have the same distribution, after n spins the standard deviation of the total result is σ√n, while the standard deviation of the average result is σ/√n.

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Calculation of probability, expected value and house edge in Roulette

Volatility

The parameters described above are theoretical; in practice, volatility is an extremely important factor to consider.
Volatility analysis is important for both operators and professional players because it helps distinguish fluctuations consistent with the mathematical model from deviations that deserve further investigation. Standard deviation does not predict the next spin: it measures the dispersion of observed results around their expected value.
To understand which fluctuations may be considered normal, a useful starting point is a common statistical measure called standard deviation.

Even without a maximum betting limit, the Martingale does not turn a negative-expected-value game into a favorable one. In the unrealistic model of unlimited capital and no limits of any kind, the probability of eventually recording a win is 1, but the required exposure grows exponentially and the expected capital required is unlimited; it is therefore incorrect to describe it as a winning strategy. In every feasible version, with finite capital or a finite number of doublings, expected value remains negative and a long unfavorable sequence can produce a very large loss.

To distinguish expected-value calculations from stake management alone, see Roulette staking plans and progressions. For criteria that select a bet according to previous results, see bet selection.

In single-zero Roulette, most bets carry a house edge of 2.70%, reduced to approximately 1.35% on even-money bets when La Partage or En Prison applies. These values are often higher than those obtainable in Baccarat or Blackjack under favorable conditions. Roulette is, however, generally slower: for the same average wager, expected hourly loss also depends on the number of spins played, the rules and the total betting volume. A complete assessment must therefore consider both the percentage edge and hourly EV.

Probabilities, payouts, house edge and standard deviation in French Roulette

Red/Black, Even/Odd, High/Low — with La Partage when zero occurs

Probability of winning: 48.6%
Probability of recovering half the stake: 2.7%
Payout: 1 to 1
House edge: 1.35%
Standard deviation: 0.99

Columns and Dozens

Probability of winning: 32.43%
Payout: 2 to 1
House edge: 2.70%
Standard deviation: 1.40

Six Lines

Probability of winning: 16.22%
Payout: 5 to 1
House edge: 2.70%
Standard deviation: 2.21

Corners

Probability of winning: 10.81%
Payout: 8 to 1
House edge: 2.70%
Standard deviation: 2.79

Streets

Probability of winning: 8.11%
Payout: 11 to 1
House edge: 2.70%
Standard deviation: 3.28

Splits

Probability of winning: 5.41%
Payout: 17 to 1
House edge: 2.70%
Standard deviation: 4.07

Straight Ups

Probability of winning: 2.70%
Payout: 35 to 1
House edge: 2.70%
Standard deviation: 5.84

American Roulette mathematics: house edge

Even-money bets (Red/Black, Even/Odd, High/Low): 5.26%
All standard bets except the First Five: 5.26%
The First Five (0, 00, 1, 2, 3): 7.89%

On even-money bets, if 0 or 00 occurs, the player loses the entire stake.

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From theory to calculation

Apply the statistical concepts with the Roulette Wheel Bias Tester: chi-square, residuals and Bonferroni screening and connect EV and standard deviation to the bankroll and risk-of-ruin calculators.