Roulette probabilities and house edge: single‑zero vs double‑zero vs triple‑zero

Why this matters

With standard payouts unchanged, the house edge equals z/(36+z) where z is the number of zero pockets: single‑zero ≈ 2.70%, double‑zero ≈ 5.26%, triple‑zero ≈ 7.69%.

How the math works: pockets, payouts and the z/(36+z) formula

A standard numbered roulette layout has 36 payout‑number pockets. Casinos pay straight‑up wins as if there were 36 non‑zero outcomes (35:1 plus return of stake). When the wheel includes z green zero pockets the true number of outcomes becomes 36 + z. If payouts remain the unchanged standard amounts, the house edge on a non‑special bet is z/(36+z).

This formula follows directly from expected‑value arithmetic: a $1 straight‑up bet returns 36/(36+z) on average, so the expected loss (house edge) equals 1 − 36/(36+z) = z/(36+z). It applies to every ordinary bet type provided the payout table is unchanged.

• Pockets = 36 + z, where z = number of zeros (1, 2, or 3).
• House edge = z/(36+z) for standard payouts.
• Formula assumes unchanged 35:1 straight‑up and other standard payouts.
• Each spin is independent; house edge is a long‑run expected loss.

Single‑zero, double‑zero and triple‑zero: the numbers

Single‑zero (European/French) wheels have 37 pockets (1–36 + 0) and a baseline house edge of 1/37 ≈ 2.7027% when no special rules apply.

Double‑zero (American) wheels have 38 pockets (1–36 + 0 and 00) and an edge of 2/38 = 1/19 ≈ 5.2632%. Triple‑zero variants (1–36 + 0, 00, 000) have 39 pockets and, with standard payouts, an edge of 3/39 = 1/13 ≈ 7.6923%.

• Single‑zero: 37 pockets → house edge ≈ 2.70%.
• Double‑zero: 38 pockets → house edge ≈ 5.26%.
• Triple‑zero: 39 pockets → house edge ≈ 7.69% (if payouts unchanged).
• Triple‑zero is less common and often proprietary or novelty.

Worked examples: per‑spin and short‑session expected loss

Example (single $1 straight‑up bet, pays 35:1): single‑zero win prob = 1/37; expected return = 36/37 ≈ $0.972973; expected loss ≈ $0.027027 per $1 bet (2.7027%).

The same $1 straight‑up on double‑zero returns 36/38 ≈ $0.947368 (loss ≈ $0.052632). On triple‑zero it returns 36/39 ≈ $0.923077 (loss ≈ $0.076923). Scale these by bet size and spins (e.g., 100 spins at $10 each gives expected losses ≈ $27.03, $52.63, $76.92 respectively).

• Single $1 bet loss: ≈ $0.02703 (single‑zero).
• Single $1 bet loss: ≈ $0.05263 (double‑zero).
• Single $1 bet loss: ≈ $0.07692 (triple‑zero).
• 100 spins × $10 example: losses ≈ $27.03 / $52.63 / $76.92.

Limitations, rule variants and responsible use

Some tables use special dealer rules that alter effective edge on certain bets. 'La partage' or 'en prison' on European/French tables halve the house edge on even‑money bets; these table‑specific rules change outcomes for affected bets but do not alter the baseline z/(36+z) unless applied.

Casinos may also offer non‑standard side bets, electronic variants, or adjusted payouts—each requires separate evaluation. Triple‑zero wheels are less standard and may use proprietary branding or variant rules; always confirm the exact table rules before play. Remember that house edge describes long‑run average loss, not a short‑term guarantee: variance is high on single‑number bets.

• En prison / la partage can reduce edge on even‑money bets (table‑specific).
• Non‑standard payouts or side bets change expected value; verify rules.
• Triple‑zero availability and exact rules vary by operator/jurisdiction.
• House edge is a long‑run expectation; short sessions vary widely.

Practical checklist

• Confirm the wheel type at the table (1‑zero, 2‑zero, or 3‑zero) before betting.
• Check table rules for 'la partage' or 'en prison' if you plan even‑money bets.
• Convert percentage house edge into dollars for your typical bet size and session length.
• Set loss limits and a fixed bet sizing rule to manage bankroll exposure.