Roulette House Edge: Formula and Calculation

The roulette house edge is the share of every dollar wagered that the casino keeps on average, and it comes from paying winners less than the true odds. On a European single-zero wheel it is 1 ÷ 37 ≈ 2.70%, and on an American double-zero wheel it is 2 ÷ 38 ≈ 5.26%. A single formula, (N − 36) ÷ N, where N is the number of pockets, gives the edge for every standard bet on the wheel.

The house edge is not the chance of losing a bet, and it is not the amount you lose in one session. It is an average percentage of money wagered, and it works only over many bets. A player can win on any one spin and the casino still earns its edge across the total of everyone’s bets.

This guide derives the roulette house edge from the difference between true odds and payout odds, shows three worked examples with every step, compares wheel types and special bets in a table, and lists the mistakes that cause the most confusion. All dollar figures are hypothetical.

True Odds Versus Payout Odds

Every house edge starts with a gap between two numbers. The true odds of a bet are the odds against it winning, which come from counting outcomes. The payout odds are what the casino actually pays. A fair game pays exactly the true odds, and the house edge is the shortfall.

A single number on a European wheel wins 1 time in 37, so the true odds against are 36 to 1. The casino pays 35 to 1. The missing 1 unit out of every 37 is the edge. On an American wheel the true odds against are 37 to 1, yet the payout is still 35 to 1, so the shortfall is 2 units out of 38.

Two related terms are worth separating. The house edge is the casino’s average gain per dollar wagered. The return to player (RTP) is the average amount returned per dollar, which is 100% minus the house edge. A 2.70% edge means a 97.30% RTP. Neither number says how likely you are to win any single bet, which is why the edge is best read alongside expected value.

The key fact for roulette is that the wheel prices every bet from the same 36 units. A bet on n numbers pays (36 ÷ n − 1) to 1, as if the wheel had only 36 pockets. The extra zero pockets are what create the edge.

The House Edge Formula

Imagine placing $1 on every one of the N pockets. You wager $N in total. Exactly one pocket wins, and it returns $36 (the 35 to 1 winnings plus the stake). The casino keeps the rest, so its average gain per dollar is the following.

where:

N is the number of pockets on the wheel: 37 for single-zero, 38 for double-zero, 39 for triple-zero.

36 is the number of units the wheel pays out, because every winning bet is priced as if the wheel had 36 numbers.

The same result comes from the expected value formula. For a bet that pays X to 1 and wins with probability p, the player’s EV per $1 is shown below, and the house edge is its negative.

where:

p is the probability the bet wins.

X is the payout odds, such as 35 for a straight-up number.

(1 − p) is the probability the bet loses.

Finally, the money-per-session view uses a rule from the expected value formula.

where:

total amount wagered adds up every bet placed, not the starting bankroll.

expected loss is an average over many bets, not a forecast of one session.

The two routes agree because they describe the same thing. With p = 1 ÷ N and X = 35, the second formula reduces to (N − 36) ÷ N for a single number, and every other roulette bet comes out the same.

Worked Examples, Step by Step

Example 1: a single number, European wheel. Suppose you bet $1 on one number on a 37-pocket wheel.

Step 1: p = 1 ÷ 37 and X = 35.

Step 2: EV = (1 ÷ 37) × 35 − (36 ÷ 37) = (35 − 36) ÷ 37 = −1 ÷ 37.

Step 3: house edge = 1 ÷ 37 ≈ 2.70%, and RTP ≈ 97.30%. The shortcut gives (37 − 36) ÷ 37, the same value.

Example 2: red, American wheel. Suppose you bet $1 on red on a 38-pocket wheel. The bet covers 18 numbers and pays 1 to 1.

Step 1: p = 18 ÷ 38 and X = 1.

Step 2: EV = (18 ÷ 38) × 1 − (20 ÷ 38) = −2 ÷ 38.

Step 3: house edge = 2 ÷ 38 ≈ 5.26%, and RTP ≈ 94.74%. The shortcut gives (38 − 36) ÷ 38, again the same.

Example 3: expected loss over 100 spins. Suppose someone bets $25 on every one of 100 spins, so the total wagered is 25 × 100 = $2,500. On a European wheel, expected loss ≈ 2,500 ÷ 37 ≈ $67.57. On an American wheel, expected loss ≈ 2,500 × 2 ÷ 38 ≈ $131.58. The same betting pattern costs about twice as much on the double-zero wheel, which is an average, not a forecast of what any 100 spins will do.

House Edge Across Wheels and Bets

The table below applies the formula and the expected value method to common cases. Each row states its rule set, since the edge depends on the wheel and on the version of the rules in use.

The la partage row works because, when the ball lands on zero, half of an even-money bet is returned. The zero pocket occurs 1 time in 37, so the average loss on that pocket drops from a full unit to half a unit: (1 ÷ 37) × 0.5 ≈ 1.35%. La partage and similar rules are not offered at every table, so check the rules before assuming it.

The five-number bet is the one common exception on an American wheel. Its payout of 6 to 1 would be fair at 6.6 to 1, so it carries an edge of 7.89% instead of 5.26%. Picking the number of pockets you cover does not lower the edge, because every other standard bet is priced from the same 36 units.

Common Mistakes and Misconceptions

The first mistake is confusing the house edge with the chance of losing. Red on a European wheel loses 19 times in 37, about 51.35%, yet its house edge is 2.70%. The two numbers answer different questions: how often you lose, and how much you lose on average per dollar.

The second is applying the percentage to the starting bankroll. The edge applies to the total amount wagered. Betting $25 on 100 spins wagers $2,500, and the expected loss is based on that figure, not on the money you brought.

The third is thinking a different bet type changes the edge. Splitting stakes between a number, a color and a dozen does not average out to something lower than 2.70% or 5.26%, because each bet carries the same edge and the average of identical edges is that edge.

Finally, a low edge is not a prediction of winning. A 2.70% edge still means the average result is a loss, and a short session can end ahead because of variance. The edge shows itself reliably only across large totals.

Quick Reference and How to Check Your Answer

Check an edge by computing it two independent ways. For a European straight-up bet, (37 − 36) ÷ 37 ≈ 2.70% should match −((1 ÷ 37) × 35 − 36 ÷ 37). Then confirm RTP plus the edge equals 100%: 97.30% + 2.70%. If the two methods disagree, recheck the number of pockets and the payout odds, since those are the usual sources of error.

Related Concepts and Next Steps

The house edge connects to the rest of gambling math through two ideas. Counting outcomes is the starting point in how probability works in gambling, and quoting prices in sports betting uses the same true-versus-payout gap, which is explained in the guide to odds conversion and the bookmaker’s margin.

The next topic after the edge is variance: the edge says what you lose on average, and variance says how far a real session can stray from that average in either direction.