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The house edge, derived from scratch

Roulette’s edge is a payout gap, not a fee: bets pay as if the wheel had 36 pockets while the ball sees 37 (or 38, or 39). Straight-up EV on a European wheel = (1/37)(35) − (36/37) = −1/37 = −2.703% — and because every standard bet’s payout is built on the same 36-pocket fiction, every bet on one wheel prices identically. Extra zeros scale it: 2/38 → 5.263%, 3/39 → 7.692%; la partage rebates half a zero: 1.351%.⚙ computed

One derivation, four wheels

WheelStraight-up EVEdge
European(1/37)(35) − 36/37 = −1/372.703%
American(1/38)(35) − 37/38 = −2/385.263%
Triple-zero(1/39)(35) − 38/39 = −3/397.692%
French, even-money(18/37)(1) + (1/37)(−½) − 18/371.351%

Edge vs hold — don’t confuse the two

Casinos report hold (share of buy-ins kept) at 15–25% for roulette — alarming next to “2.7%” until you see the mechanism: players recycle winnings, so an evening’s wagers total several times the buy-in, and 2.7% of the big number equals ~20% of the small one. Same math, two denominators. Our simulator’s meter tracks the honest one: edge on wagers.

Why no bet escapes

Combination bets are sums of standard bets, so their EVs sum too — no lattice of chips can average negative parts into a positive whole. The lone standard exception proves the rule by being worse: the American basket breaks the 36-pocket fiction in the house’s favor at 7.89%. Everything else you can do with chips is choosing variance — the SD column — at a fixed price.

Related

How other games compareOur validation of these numbersWhy systems can’t dent itTurning edge into $/hour

Frequently asked

What is the house edge in roulette?
2.70% European, 5.26% American, 7.69% triple-zero — on every standard bet — and 1.35% on French even-money with la partage. It arises from paying 35:1 on odds that are truly 36:1 or worse.
Why is the house edge the same for every bet?
All payouts are scaled to a fictional 36-pocket wheel: n covered numbers pay (36−n)/n. The gap between 36 and the real pocket count is constant, so the percentage is too — our engine verifies it to the third decimal per bet.
Can the house edge be beaten?
Not by staking patterns — EVs add. Historical exceptions attacked physics (biased wheels, dealer signatures), not math, and modern maintenance targets exactly those.