The arithmetic that fixes every roulette outcome before the ball is spun.
House edge, the actual maths
Annabel Cavendish
Editor · 14 May 2026
The Elegant and Maddening Fact
Start with the arithmetic, because it's cleaner than most people realise.
A European wheel has 37 pockets. When you bet on a single number, it pays 35 to 1. The fair payout would be 36 to 1, because there are 36 other outcomes. The casino keeps one pocket's worth of your money, which is 1/37 of everything wagered. In percentage terms, that is 2.703%. The decimal never terminates. The casino rounds payouts down, always, never up, and that rounding is the entire source of the house advantage.
According to our roulette basics analysis, every single bet on the European table, all 35 of them, carries exactly 2.703% house edge. Straight-ups, splits, streets, corners, columns, dozens, red and black, odd and even, high and low. Every single one. This isn't a coincidence; it's an algebraic identity. The payout for a bet covering K numbers is (37-K)/K - 1 = (36-K)/K, and subtracting the one missing pocket's cost always resolves to -1/37, regardless of how many numbers you cover.
There is exactly one exception in standard roulette, and it lives on the American wheel.
The One Bet You Should Never Touch
The basket bet on an American double-zero wheel covers zero, double-zero, 1, 2, and 3. It pays 6 to 1. The problem: five numbers in 38 pockets gives a mathematically fair payout of 6.2 to 1. The casino rounds down to 6, creating a house edge of 7.895%. Every other American wheel bet carries 5.26%. The basket bet carries 7.89%. The reason is arithmetic: five doesn't divide cleanly into 38, so the standard payout formula produces a rounding error that doesn't appear anywhere else on the table. we confirm it's the only payout-schedule rounding error in standard roulette. Simply don't play it.
Variance: The Ride, Not the Edge
Here is where most conversations about roulette bet types go wrong.
The universal advice is to stick to even-money outside bets because they are "safer." This advice is imprecise. The house edge is identical regardless of bet type on a European wheel. What differs between a straight-up bet and a red-or-black bet is standard deviation: the width of the range of outcomes you might experience in a given session.
The formula for standard deviation per spin is (payout + 1) multiplied by the square root of the probability times (1 minus the probability). For a straight-up bet paying 35 to 1 with a 1/37 probability, that comes to approximately 5.84 units per spin. For a red or black bet paying 1 to 1 with an 18/37 probability, it's approximately 0.999 units per spin. Same expected loss. About six times more volatility.
What that means practically: over 200 spins at £100 flat on red, your expected loss is £541 and the 95% confidence interval runs from roughly -£3,311 to +£2,230. Over 200 spins at £100 flat on a single straight-up number, the expected loss is still £541, but the 95% interval runs from roughly -£16,722 to +£15,641. Same edge. About twelve times the range of outcomes.
If your goal is session longevity on a fixed budget, outside bets deliver that. If your goal is a meaningful win against a meaningful stake, inside bets are the mechanism. The house doesn't give you better odds for choosing one over the other. The edge is the same. The ride is not.
What It Costs Per Hour
Round numbers are more useful than percentages when you're planning an evening.
According to Roulette77's spins-per-hour analysis, a live London table runs approximately 30 spins per hour. At that pace, flat betting even-money at £100 per spin, your expected loss is approximately £81 per hour. At £500 flat, it's £405 per hour. At £5,000 flat, it's £4,054 per hour. These figures assume a European single-zero wheel.
On an American wheel, those numbers multiply by roughly 1.95. A four-hour session at £5,000 flat on a European wheel costs you on average around £16,200 in expected losses. The same session on an American wheel costs around £31,600. Les Ambassadeurs at Hamilton Place runs American roulette. The "same game" framing conceals a very significant difference at those stakes.
The one category where this calculus shifts is La Partage: on even-money bets at a French table with La Partage, the edge drops to 1.35% and the hourly cost at £5,000 flat falls to approximately £2,025. That's a meaningful difference across any serious session.
The Eudaemons and the Double-Sided Ledger
There is a footnote to all of this that I find instructive, not because it's actionable but because it demonstrates how precise the math is.
In the late 1970s, Doyne Farmer and Norman Packard, physics graduate students at UC Santa Cruz who would later become founding figures of chaos theory, built a wearable computer to predict roulette outcomes by timing the ball's revolution. As Farmer recounts on his faculty website, they achieved a 20% positive edge over the house on the bets they made using the computer's guidance. And they made almost no money. The reason: every pound not bet using the computer's prediction was still subject to the 2.703% drain. Cover bets, misdirection wagers, play before the device was ready, all of it was feeding the house edge at the standard rate. The physics edge was real. The accounting was brutal. The maths worked perfectly on both sides of the ledger.
Rebate Programmes: The Arithmetic of Almost
If you're ever offered a rebate programme by a casino host, here's the arithmetic.
High-roller rebate or dead-chip programmes return a percentage of your losses as non-negotiable chips, which can be wagered but not cashed directly. If the rebate percentage exceeds the house edge on roulette, you have a positive-expectation game. That percentage is 2.703% on European roulette. Documented rebate programmes run to perhaps 1.6% at the very top tier. So even the most generous known programme doesn't get you to even on roulette. Baccarat banker at higher rebate percentages is a different calculation, but that is another lesson.
What the edge gives the casino is certainty over time. What it gives you, in the short term, is a range of outcomes wide enough to be entertaining and occasionally wonderful. Just don't confuse the entertainment with an investment thesis.
Key numbers
Bet type
Pays
House edge
Std deviation per unit
Straight-up (1 number)
35:1
2.703%
5.84
Split (2 numbers)
17:1
2.703%
~4.11
Street (3 numbers)
11:1
2.703%
~3.35
Corner (4 numbers)
8:1
2.703%
~2.88
Six-line (6 numbers)
5:1
2.703%
~2.27
Column or Dozen
2:1
2.703%
~1.45
Red/Black, Odd/Even, High/Low
1:1
2.703%
~1.00
Even-money with La Partage
1:1
1.351%
~1.00
American (all standard bets)
varies
5.263%
higher
American five-number basket
6:1
7.895%
n/a
Bet level
Expected loss per hour (European, 30 spins, even-money)
£100 flat
£81
£500 flat
£405
£1,000 flat
£811
£5,000 flat
£4,054
Sources: our in-house analysis; Roulette77 spins per hour
Annabel
0:000:00
Welcome to the lesson on the house edge.
Today we're doing the maths.
I'm Annabel, and I want to talk to you about the house edge, which is a phrase that gets thrown around constantly at roulette tables and understood, at a fundamental level, by almost nobody sitting at them.
By the end of this lesson you will understand it precisely, and more usefully, you'll understand what it costs you in pounds per hour at various bet sizes, which is the number that actually matters.
Let's start with European roulette.
There are thirty-seven numbers on the wheel, zero through thirty-six.
When you bet on a single number, it pays thirty-five to one.
The fair payout would be thirty-six to one, because there are thirty-six other outcomes, and the casino keeps one pocket's worth of your money.
That fraction is one thirty-seventh of everything wagered.
In percentage terms, that is two point seven zero three percent.
The decimal never terminates.
The casino rounds payouts down, always, never up, and that rounding is the source of the entire house advantage.
Here is the elegant and slightly maddening fact about European roulette: every single bet on the table, all thirty-five of them, has exactly the same house edge of two point seven percent.
Straight-ups, splits, streets, corners, columns, dozens, red and black, odd and even, high and low.
Every single one.
This is not a coincidence but an algebraic identity baked into the payout structure.
There is exactly one exception on any standard roulette table in the world, and it's on the American wheel.
The five-number basket bet, covering zero, double-zero, one, two, and three, pays six to one.
The fair payout would be six point two to one, because five numbers in thirty-eight pockets gives you a slightly different ratio.
Six doesn't divide neatly into thirty-six plus two, so the casino rounds it down to six, creating a house edge of seven point eight nine five percent on that specific bet.
It's the only payout-schedule rounding error in standard roulette.
Every other American wheel bet costs you five point two six percent.
The basket bet costs you seven point eight nine.
Simply don't play it.
Now.
Standard deviation.
This is where most conversations about roulette bet types get it wrong.
The universal advice is to stick to even-money outside bets because they are "safer." This advice is imprecise.
The house edge is identical regardless of bet type on a European wheel.
What differs between a straight-up bet and a red-or-black bet is standard deviation, which is the width of the range of outcomes you might experience in a given session.
On a straight-up bet, the standard deviation per spin is approximately five point eight four times your stake.
On red and black, it's approximately zero point nine nine nine times your stake.
Same expected loss.
About six times more volatility.
If your goal is session longevity on a fixed budget, outside bets give you that.
If your goal is a meaningful win against a meaningful stake, inside bets are the mechanism.
The house doesn't give you better odds for choosing one over the other.
The edge is the same.
The ride is not.
Let's make this concrete with some hourly numbers, because I think round numbers are more useful than percentages when you're planning an evening.
At a London table running thirty spins per hour, flat betting even-money at one hundred pounds per spin, your expected loss is approximately eighty-one pounds per hour.
At five hundred pounds flat, it's four hundred and five pounds an hour.
At five thousand pounds flat, it's four thousand and fifty-four pounds an hour.
That last number is the one worth sitting with.
A four-hour session at five thousand pounds flat on a European wheel costs you on average sixteen thousand and something in expected losses.
The same session at an American wheel, which Les Ambassadeurs runs, multiplies that by just under two.
The "same game" framing conceals a very significant difference at those stakes.
There is a footnote to all of this that involves the Eudaemons, a group of physics graduate students at UC Santa Cruz in the late nineteen seventies who built a wearable computer to predict roulette outcomes by timing the ball's revolution.
They achieved, in Doyne Farmer's own words, a twenty percent positive edge over the house on the bets they made using the computer's guidance.
And they made almost no money.
The reason is instructive: every pound not bet using the computer's prediction was still subject to the two point seven percent drain.
Cover bets, misdirection wagers, play before the device was ready, all of it was feeding the house edge at the standard rate.
The physics edge was real.
The accounting was brutal.
The maths worked perfectly on both sides of the ledger.
One more thing that is worth knowing if you are ever offered a rebate programme by a casino host.
High-roller rebate or dead-chip programmes return a percentage of your losses as non-negotiable chips, which can be wagered but not cashed directly.
If the rebate percentage exceeds the house edge on roulette, you have a positive-expectation game.
That percentage is two point seven percent on European roulette.
Documented rebate programmes run to perhaps one point six percent at the very top tier.
So even the most generous known programme doesn't get you to even on roulette.
Baccarat banker at higher rebate percentages is a different calculation, but that is another lesson entirely.
What the edge gives the casino is certainty over time.
What it gives you, in the short term, is a range of outcomes wide enough to be entertaining and occasionally wonderful.
Just don't confuse the entertainment with an investment thesis.
