European vs American vs French Roulette: Deriving Every House Edge from First Principles

For a quick comparison of house edges by wheel type, see Roulette House Edge by Wheel. This lesson derives those numbers from first principles.

Where Does the House Edge Actually Come From?

Every roulette payout is calculated as if the zero pockets don't exist — as if the wheel has exactly 36 numbers. But the zero does exist. That mismatch between the payout structure and the actual probability is the house edge, and it can be derived in a few lines of arithmetic for every variant.

Understanding the derivation means you can calculate the edge on any hypothetical wheel, not just memorise a table.

How Do You Derive the European House Edge?

Take any bet. We'll use red/black because the numbers are clean, but the result generalises to every standard bet.

Setup: European wheel, 37 pockets — 18 red, 18 black, 1 green (zero). A $1 bet on red pays 1:1.

Step 1 — Expected value (EV):

EV = (probability of winning × net gain) + (probability of losing × net loss)

EV = (18/37 × $1) + (19/37 × −$1)

EV = $0.4865 − $0.5135

EV = −$0.0270

Step 2 — Express as a percentage of the wager:

House edge = |EV| / wager = $0.027 / $1 = 2.70%

Does this change for inside bets?

No. Take a straight-up bet (single number, pays 35:1):

EV = (1/37 × $35) + (36/37 × −$1)

EV = $0.9459 − $0.9730

EV = −$0.0270

The edge is identical. This holds for every standard bet because every payout is computed from the same formula: (36 ÷ numbers covered) − 1. The zero is excluded from the payout calculation but included in the probability calculation. That consistent asymmetry produces 2.70% across the board.

How Do You Derive the American House Edge?

The American wheel has 38 pockets: 18 red, 18 black, 2 green (0 and 00). Payouts remain on the 36-number scale.

Red/black, $1 bet:

EV = (18/38 × $1) + (20/38 × −$1)

EV = $0.4737 − $0.5263

EV = −$0.0526 → 5.26%

Straight-up, $1 bet:

EV = (1/38 × $35) + (37/38 × −$1)

EV = $0.9211 − $0.9737

EV = −$0.0526 → 5.26%

Again, the same edge for every bet — with one exception.

The basket bet exception

The five-number "basket" (or top-line) bet covers 0, 00, 1, 2, 3 and pays 6:1.

EV = (5/38 × $6) + (33/38 × −$1)

EV = $0.7895 − $0.8684

EV = −$0.0789 → 7.89%

This is the only standard roulette bet where the edge diverges from the wheel's baseline. It exists because 6:1 is not the correct 36-number payout for a 5-number bet (which would be 36/5 − 1 = 6.2:1, a non-integer). The casino rounds down to 6:1, and you pay the difference. Avoid this bet entirely.

How Do You Derive the Triple-Zero Edge?

Triple-zero wheels (0, 00, 000) have 39 pockets. They appeared in Las Vegas around 2016 and have spread to lower-minimum tables.

EV (red/black) = (18/39 × $1) + (21/39 × −$1) = −$0.0769

House edge = 7.69%

For context, that's nearly three times the European edge and worse than most slot machines. The derivation pattern is clear: each added zero pocket increases the denominator by 1 while the payout numerator (36) stays fixed.

The general formula: house edge = z / (36 + z), where z is the number of zero pockets (Ethier, S.N., The Doctrine of Chances, Cambridge University Press, 2010; accessed August 2026).

How Does La Partage Work — and Why Does It Cut the Edge in Half?

La partage ("the sharing") is a French rule that applies only to even-money bets (red/black, odd/even, high/low). When zero lands:

1. Even-money bets lose only half the stake instead of the full amount.
2. The other half is returned to the player.
3. All other bet types lose normally.

Full EV derivation with la partage

$1 bet on red, European wheel (37 pockets):

• Ball lands red (18/37): win $1 → net +$1
• Ball lands black (18/37): lose $1 → net −$1
• Ball lands zero (1/37): lose half → net −$0.50

EV = (18/37 × $1) + (18/37 × −$1) + (1/37 × −$0.50)

EV = $0.4865 − $0.4865 − $0.0135

EV = −$0.0135 → 1.35%

That's exactly half the standard European edge. The red and black outcomes cancel; the entire house edge comes from the half-unit loss on zero.

How Does En Prison Differ from La Partage?

En prison ("in prison") is an alternative French rule. When zero lands on an even-money bet:

1. Your bet is not collected. Instead, the croupier places a marker on it — it's "in prison."
2. The next spin decides its fate: if your original bet wins, the stake is returned (but you don't win anything extra). If it loses, the house takes it. If zero hits again, the bet stays imprisoned for another spin (in some casinos, it's forfeited on the second zero).

EV derivation for en prison (single-imprisonment model)

$1 bet on red, zero hits → bet imprisoned. Next spin outcomes:

• Red (18/37): you get your $1 back → net $0
• Black (18/37): you lose $1 → net −$1
• Zero again (1/37): you lose $1 → net −$1 (forfeiture rule)

Expected loss from the imprisonment event = (18/37 × $0) + (19/37 × $1) = $0.5135

But the imprisonment only happens with probability 1/37, so:

Full EV = (18/37 × $1) + (18/37 × −$1) + (1/37 × −$0.5135)

EV = −$0.0139 → 1.39%

Under the stricter forfeiture rule (double-zero = lost), en prison is very slightly worse than la partage (1.39% vs 1.35%). Under the lenient rule (stay imprisoned on second zero), it converges to exactly 1.35% over infinite horizons. In practice, the difference is negligible (Hannum, R.C. & Cabot, A.N., Practical Casino Math, 2005; accessed August 2026).

La partage vs en prison: practical comparison

Which Platforms Actually Offer French Rules?

This is where theory meets reality. Many online casinos offer a game titled "French Roulette" that uses the French table layout (printed neighbour bets, racetrack) but does not apply la partage or en prison. The label is cosmetic.

To verify, check the game's info/help screen for explicit mention of the rule, or place an even-money bet and wait for zero to hit. If you lose the full amount, the game doesn't apply French rules regardless of its name.

Providers known to offer genuine la partage:

Availability varies by jurisdiction and platform. On wild.io, check the game info panel before committing to a session — the specific rules are listed there.

How Did Each Variant Develop Historically?

Roulette's evolution explains why three variants coexist:

• French roulette (1796): The original. Blaise Pascal's perpetual-motion experiments led to the wheel; the Blanc brothers later removed the double zero in 1843 to compete with rival casinos in Bad Homburg, Germany. La partage and en prison were concessions to attract high-rolling players (Mézrich, B., Busting Vegas, 2005; accessed August 2026).
• European roulette: The single-zero standard that spread across the continent from the Blanc family's Monte Carlo casino. Functionally identical to French roulette without the partage/prison rules.
• American roulette: Early American casinos reinstated the double zero to increase profits. The different number sequence evolved independently as American wheel manufacturers diverged from European designs.

Which Variant Should You Play?

The mathematics are unambiguous:

1. French roulette with la partage (1.35% on even-money bets) if available.
2. European roulette (2.70%) as the standard choice.
3. American roulette (5.26%) only when no single-zero option exists.
4. Triple-zero — never, under any circumstances.

For a side-by-side reference of these edges, see Roulette House Edge by Wheel. To understand how these edges interact with specific bet types and bankroll variance, continue to the next lesson: Roulette Bets, Payouts and Probability.