Roulette Odds and House Edge: A Derivation

The house edge in roulette is not a number that can be quoted without the wheel. A single-zero wheel has 37 pockets. A double-zero wheel has 38. Both pay a winning straight-up number at 35 to 1. That difference of one pocket is the entire reason the two games cost different amounts to play, and it is the first thing to establish before any figure is stated.

The wheel determines the arithmetic

A roulette wheel is a physical device with a fixed number of equally likely outcomes. On a European wheel, the pockets are numbered 1 through 36, plus a single zero. That is 37 pockets. On an American wheel, the pockets are 1 through 36, plus a zero and a double zero. That is 38 pockets. The ball lands in one pocket per spin. No pocket is more likely than another, and the previous result has no influence on the next.

The payout table is the second input. A straight-up bet on a single number pays 35 to 1. That means a winning 1 unit stake returns 35 units of profit plus the original 1 unit stake, for a total of 36 units back. The payout is 35 to 1, not 36 to 1. The difference between the payout and the true odds is the house edge.

Deriving the edge on a single-zero wheel

Take a 1 unit straight-up bet on a single number on a European wheel.

• There are 37 equally likely outcomes.
• One outcome wins. The bet returns 36 units total (35 profit plus the 1 unit stake).
• 36 outcomes lose. The bet returns 0.

The expected return is the sum of each outcome multiplied by its probability.

Expected return = (1/37 × 36) + (36/37 × 0)
Expected return = 36/37
Expected return = 0.972973

The expected loss per 1 unit staked is 1 − 0.972973 = 0.027027, or 2.70%.

Equivalently, the house edge is 1/37 = 0.027027. The single zero is the one pocket that pays nothing to the straight-up bettor but still takes up a slot on the wheel. That is the source of the edge.

Deriving the edge on a double-zero wheel

Now take the same 1 unit straight-up bet on an American wheel.

• There are 38 equally likely outcomes.
• One outcome wins. The bet returns 36 units total.
• 37 outcomes lose. The bet returns 0.

Expected return = (1/38 × 36) + (37/38 × 0)
Expected return = 36/38
Expected return = 0.947368

Expected loss per 1 unit staked = 1 − 0.947368 = 0.052632, or 5.26%.

The house edge is 2/38 = 0.052632. The double zero adds a second pocket that pays nothing on this bet, and the edge nearly doubles.

Every standard bet carries the same edge

The straight-up bet is not special. On a given wheel, every bet on the layout has the same house edge, because every bet is priced off the same 37 or 38 pockets. The payout is always set so that the ratio of winning pockets to losing pockets is one unit short of fair.

Consider a split bet on a European wheel. A split covers two numbers. The payout is 17 to 1. A winning 1 unit bet returns 18 units total (17 profit plus stake).

• Winning outcomes: 2 pockets.
• Losing outcomes: 35 pockets.

Expected return = (2/37 × 18) + (35/37 × 0)
Expected return = 36/37
Expected return = 0.972973

Same 2.70% edge. The same holds for a corner bet (4 numbers, 8 to 1), a street (3 numbers, 11 to 1), a column (12 numbers, 2 to 1), and an even-money bet like red or black (18 numbers, 1 to 1). Each pays one unit less than the true odds implied by the 37 pockets.

On the double-zero wheel, the same bets carry 5.26%, with one exception.

The five-number bet on the American wheel

The American wheel has a bet that does not exist on the European layout: the five-number bet, covering 0, 00, 1, 2, and 3. It pays 6 to 1. A winning 1 unit bet returns 7 units total (6 profit plus stake).

• Winning outcomes: 5 pockets.
• Losing outcomes: 33 pockets.

Expected return = (5/38 × 7) + (33/38 × 0)
Expected return = 35/38
Expected return = 0.921053

Expected loss per 1 unit staked = 1 − 0.921053 = 0.078947, or 7.89%.

The five-number bet is worse than every other bet on the same wheel. The reason is that the payout of 6 to 1 is not the one-unit-short-of-fair price that the other bets use. Fair odds for 5 winning pockets out of 38 would be 33 to 5, or 6.6 to 1. The casino pays 6 to 1. The shortfall is larger than on the other bets, and the edge rises to 7.89%. This bet is the only standard wager on either wheel that carries a different edge from the rest of the table.

What the edge costs per hour

A percentage of turnover is not a quantity most people can feel. The same 2.70% edge costs very different amounts depending on stake size and how many spins are played per hour.

Assume a player stakes 1 unit per spin on a single-zero wheel and plays 200 spins per hour. This is a plausible rate for a live roulette table with a dealer, though actual rates vary with the table and the player.

• Total amount staked in an hour: 1 × 200 = 200 units.
• Expected loss: 200 × 0.027027 = 5.4054 units.

At 1 unit per spin and 200 spins per hour, the expected cost is about 5.40 units per hour on a single-zero wheel.

On a double-zero wheel at the same stake and rate:

• Total staked: 200 units.
• Expected loss: 200 × 0.052632 = 10.5264 units.

About 10.53 units per hour. The double-zero wheel costs roughly 5.13 units more per hour at this stake and rate.

If the stake is 5 units per spin instead of 1, multiply both figures by 5: about 27.03 units per hour on single-zero, about 52.63 units per hour on double-zero. If the rate is 100 spins per hour instead of 200, halve them. The edge percentage does not change; the money does.

What the figures depend on

Every number above depends on specific conditions. The table below states them.

A figure for “roulette” without the wheel type is incomplete. The same game name covers both wheels, and the edge differs by a factor of nearly two. The five-number bet is a separate row because it is the one bet on the American layout that does not match the rest of the table.

What people get wrong

The most common error is treating the wheel as having a memory. A number that has not appeared for many spins is not more likely to appear on the next spin. Each spin is independent. The probability of any specific number on a single-zero wheel is 1/37 on every spin, regardless of what came before. The same applies to red or black, odd or even, and every other bet. There is no mechanism by which past results change the physical probabilities of the next result. The wheel does not know what it has done.

This mistake is natural because people look for patterns in sequences. A run of ten reds feels like it must be followed by black, but the wheel has no state to correct. The probability of black on the next spin is 18/37 on a single-zero wheel, the same as it was on the first spin. The expected loss per spin is unchanged.

The second common error is believing that a staking system changes the edge. Martingale, d’Alembert, Fibonacci, and Labouchere are progressions that change the size of bets after wins or losses. They do not change the payout table or the number of pockets. The expected value of each individual bet is the same regardless of how the stake was chosen. A progression reshapes the distribution of outcomes: it makes many small wins more likely and a few large losses less likely, or the reverse. The mean of the distribution is fixed by the wheel and the payout. No sequence of bet sizes can move it. A player using Martingale on a single-zero wheel still faces a 2.70% expected loss per unit staked. The system changes the variance, not the edge.

This error is natural because progressions produce frequent small wins, which feel like evidence of an edge. The occasional large loss that balances the books is easy to treat as an outlier rather than the other side of the same distribution.

A third error is comparing games by their best-case session rather than their expected cost. A player who wins a session on a double-zero wheel has not beaten the 5.26% edge; they have experienced one draw from a distribution whose mean is negative. Over many spins, the average converges toward the expected loss. The edge is a long-run property, and the long run is longer than most sessions.

Live roulette and online roulette

The arithmetic above applies to any roulette wheel with the stated number of pockets and the stated payout. A live roulette table streamed to a screen uses a physical wheel, and the pocket count and payout table determine the edge in the same way. An online roulette game that uses a random number generator to simulate a wheel follows the same payout structure, and the edge is set by the same relationship between outcomes and payouts. The version matters: a game labelled “European roulette” should have 37 pockets and a 2.70% edge on standard bets; a game labelled “American roulette” should have 38 pockets and a 5.26% edge. If a game uses a different payout table, the edge changes, and the derivation must be redone with the actual numbers.

The rule for any roulette figure is the same as for any casino figure: state the wheel, the payout, and the bet. Without those, the percentage is not a fact about the game; it is a number without a source.

FAQ

What is the house edge in roulette? On a single-zero wheel with standard payouts, the house edge is 2.70% on every bet except none, because all standard bets are priced off 37 pockets. On a double-zero wheel, it is 5.26% on every bet except the five-number bet, which is 7.89%. The edge depends on the wheel, not on the bet size or the pattern of previous spins.

Is European roulette better than American roulette? European roulette has one zero and 37 pockets, giving a 2.70% edge. American roulette has two zeros and 38 pockets, giving a 5.26% edge on standard bets. At the same stake and spin rate, the American wheel costs nearly twice as much per hour in expected terms. The difference is the extra pocket, not the layout or the dealer.

Does a betting system reduce the house edge in roulette? No. Martingale, d’Alembert, Fibonacci, and Labouchere change the size and timing of bets. They do not change the payout table or the number of pockets. The expected loss per unit staked remains 2.70% on a single-zero wheel and 5.26% on a double-zero wheel. A system changes the distribution of outcomes, not the mean.

How much does roulette cost per hour? It depends on the wheel, the stake, and the number of spins per hour. At 1 unit per spin and 200 spins per hour, the expected cost is about 5.40 units per hour on a single-zero wheel and about 10.53 units per hour on a double-zero wheel. At 5 units per spin, multiply by five. The edge percentage stays the same; the money changes with stake and rate.

Can a number be due on a roulette wheel? No. Each spin is independent. The probability of any number on a single-zero wheel is 1/37 on every spin, regardless of what has appeared before. The wheel has no memory and no mechanism to correct a sequence. The expected loss per spin is the same after a long run of one colour as it was on the first spin.