Roulette house edge, European vs American: every dollar costs the same
The house edge in roulette is 2.70% on every bet of a European single-zero wheel and 5.26% on an American double-zero wheel, except the American five-number top line, which costs 7.89%. With la partage (French rules) the even-money bets cost 1.35%. In money: $25 a spin on red for 500 spins averaged -$338.21 on the European wheel and -$653.14 on the American one in 10,000 simulated sessions. No way of spreading or sizing the bets changes the edge.
Every dollar you bet costs 2.7 cents on average, no matter how you bet it: spreading $25 across the table only changes how far a session swings.
Chapter 1 of the spinner roulette research (chapters 0 to 9): Every dollar costs the same.
Three players bet $25 a spin for 500 spins from $5,000. One puts it all on red, one puts $12 on red and $13 on the third dozen, one puts $1 on each of 25 single numbers. On average all three lose the same: −$338.21, −$336.64 and −$337.98, where the wheel predicts −$337.84.
What changes is how far a session swings. The typical swing shrinks from $557 to $490 to $375 as the bet covers more numbers, and the worst 5 % of sessions improve from −$1,480 to −$1,129. Smaller swings cut both ways: the chance of ending ahead falls from 25.5 % to 17.5 %.
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The numbers
| Bet | Average | Typical | 10th pct | 90th pct | SD | Worst 5 % | Ending ahead |
|---|---|---|---|---|---|---|---|
| Red | -$338.21 | -$350 | -$1,050 | +$400 | $557.44 | -$1,479.60 | 25.5% |
| Red + a dozen | -$336.64 | -$338 | -$953 | +$289 | $489.79 | -$1,354.56 | 24.5% |
| 25 numbers | -$337.98 | -$332 | -$800 | +$136 | $374.57 | -$1,128.82 | 17.5% |
Takeaways
- Same average loss whatever the spread: about $338 per 500 spins at $25 a spin.
- Covering more numbers narrows the swings: the worst sessions are milder and the winning ones rarer.
- Spreading is a comfort choice, not an edge.
European vs American roulette: the house edge of every bet
Each bet pays a little less than its true odds because of the zero pockets. The edge is one minus the payout times the chance of winning: on a single-zero wheel every bet works out to the same share; the extra zero of the American wheel raises it, and the five-number top line is worse still.
| Bet | Pays | European (single zero) | American (double zero) | Triple zero |
|---|---|---|---|---|
| Straight up (one number) | 35 to 1 | 2.70% | 5.26% | 7.69% |
| Split (two numbers) | 17 to 1 | 2.70% | 5.26% | 7.69% |
| Street (three numbers) | 11 to 1 | 2.70% | 5.26% | 7.69% |
| Corner (four numbers) | 8 to 1 | 2.70% | 5.26% | 7.69% |
| Six-line (six numbers) | 5 to 1 | 2.70% | 5.26% | 7.69% |
| Dozen or column (12 numbers) | 2 to 1 | 2.70% | 5.26% | 7.69% |
| Red/black, odd/even, 1-18/19-36 | 1 to 1 | 2.70% | 5.26% | 7.69% |
| Basket: first four / top line with the zeros | 8 / 6 / 5 to 1 | 2.70% | 7.89% | 7.69% |
La partage (French rules, single zero only): when the ball lands on zero, even-money bets lose only half, so their edge falls to 1.35%. All other bets keep the full edge.
What the extra zero costs in a session
| Wheel | Average result | Typical session | Chance of ending ahead | Realized edge | Wagered per session |
|---|---|---|---|---|---|
| European (single zero) | -$338.21 | -$350 | 25.5% | 2.71% | $12,500 |
| American (double zero) | -$653.14 | -$650 | 11.4% | 5.22% | $12,500 |
The math
Covering c numbers with $1 wins with probability c/37 and returns 36/c dollars on a hit, so the expectation per dollar is (c/37)(36/c) − 1 = −1/37 for every c. Spread changes only the variance per dollar, 36²/(37c) − (36/37)²: 1.00 for red (c = 18) and 0.45 for 25 straight-ups.
Over n independent spins the session standard deviation is √n times the per-spin one. At $25 and n = 500: red $559 predicted, $557 measured; red + dozen $493 predicted, $490 measured; 25 straight-ups $377 predicted, $375 measured. Realized edge −2.71 %, −2.69 %, −2.70 %. CVaR 5 %: −$1,480, −$1,355, −$1,129. Medians: −$350, −$338, −$332.
EV per $1 = (c/37) × (36/c) − 1 = −1/37 ≈ −0.0270
mean(500 spins at $25) = −$12,500 / 37 = −$337.84
SD(session) = $25 × √500 × √(36²/(37c) − (36/37)²)