Roulette house edge: where 2.70% and 5.26% come from
Every bet on a European wheel pays as if there were 36 pockets when there are 37, so each one returns −1/37 of its stake on average: a 2.70% house edge. The American wheel's second zero raises it to 2/38, or 5.26%. No betting system changes it.
Where 2.70% comes from
Put $1 on every one of the 37 numbers of a European wheel. One of them wins and pays 35 to 1, so you get $36 back: the $35 win plus the $1 stake on the winner. You staked $37. You lose $1 per $37 wagered, every spin, whatever comes up. That is 1/37 = 2.70%.
Every other bet is built from the same arithmetic. A bet covering k numbers pays 36/k − 1 to 1: a split (2 numbers) pays 17, a street (3) pays 11, a dozen (12) pays 2, red (18) pays 1. The house prices each bet for a 36-pocket wheel and spins a 37-pocket one. So every bet on the layout carries the same −1/37.
| Bet | Pays | Fair payout, EU | Edge EU | Fair payout, US | Edge US |
|---|---|---|---|---|---|
| Straight up | 35:1 | 36:1 | 2.70% | 37:1 | 5.26% |
| Split | 17:1 | 17.50:1 | 2.70% | 18:1 | 5.26% |
| Street | 11:1 | 11.33:1 | 2.70% | 11.67:1 | 5.26% |
| Corner | 8:1 | 8.25:1 | 2.70% | 8.50:1 | 5.26% |
| Six line | 5:1 | 5.17:1 | 2.70% | 5.33:1 | 5.26% |
| Dozen / column | 2:1 | 2.08:1 | 2.70% | 2.17:1 | 5.26% |
| Red / black, odd / even, 1–18 / 19–36 | 1:1 | 1.06:1 | 2.70% | 1.11:1 | 5.26% |
The second zero, and the one bad bet
An American wheel adds 00 and keeps the same payouts. $1 on all 38 numbers returns $36, so the loss is $2 per $38: 2/38 = 5.26%, almost exactly double. The top line (0, 00, 1, 2, 3) is worse still. The 36-pocket pricing behind every other payout would pay five numbers 36/5 − 1 = 6.2 to 1; the house rounds it down to 6, on top of the two zeros. Its edge is 3/38 = 7.89%, the worst bet on either layout.
Going the other way, some single-zero tables use the French la partage rule: when zero comes up, even-money bets lose only half. That halves the edge on those bets to 1/74 = 1.35%. The simulator plays standard European and American rules; la partage is listed here for the arithmetic, not as something you can test on this site yet.
Expected loss is edge × total wagered
The edge applies to every dollar that goes on the table, not to your bankroll. Flat $10 bets on red:
| Session | Wagered | Expected loss, EU | Expected loss, US |
|---|---|---|---|
| 100 spins | $1,000 | $27.03 | $52.63 |
| 1,000 spins | $10,000 | $270.27 | $526.32 |
| 10,000 spins | $100,000 | $2,702.70 | $5,263.16 |
This is why betting systems can’t change the outcome. A progression changes how much you wager and when. Each of those dollars still loses 1/37 on average. The systems comparison shows 20 of them landing between −$2.52 and −$2.87 per $100 wagered on a European wheel, which is −$2.70 plus sampling noise.
When the edge overtakes luck
Over n spins the expected loss grows like n, but luck (the standard deviation of the total) only grows like √n. They cross at n = (σ / edge)². Before that point a session is mostly luck; after it, mostly edge. High-payout bets swing harder, so they hide the edge for longer. They don’t remove it.
| Bet | σ per $1, EU | Crossover, EU | σ per $1, US | Crossover, US |
|---|---|---|---|---|
| Straight up | $5.838 | 46,656 spins | $5.763 | 11,988 spins |
| Split | $4.070 | 22,680 spins | $4.019 | 5,832 spins |
| Street | $3.276 | 14,688 spins | $3.236 | 3,780 spins |
| Corner | $2.795 | 10,692 spins | $2.762 | 2,754 spins |
| Six line | $2.212 | 6,696 spins | $2.188 | 1,728 spins |
| Dozen / column | $1.404 | 2,700 spins | $1.394 | 702 spins |
| Red / black, odd / even, 1–18 / 19–36 | $1.000 | 1,368 spins | $0.999 | 360 spins |
The American crossover comes about four times sooner: double the edge, squared. What this means for your odds of finishing a session ahead is worked out exactly on chances of winning after n spins.
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- Study: European versus American roulette, the risk distribution