House edge is one formula applied to every bet on the floor. We define it, then compute it for roulette, craps, baccarat and blackjack.
I. The one formula
Every house-edge figure on every casino table comes from the same definition. The expected value (EV) of a one-unit bet is the sum, over all outcomes, of each outcome's net return times its probability. The house edge is simply the negative of that, expressed as a percentage of the stake.
EV = Σ (net payout of outcome × probability of outcome)
house edge = −EV ÷ (amount staked)
If a $100 bet has an expected value of −$1.41, the house edge is 1.41%. That is the whole of it — the rest is arithmetic for each game.II. Roulette, derived exactly
Roulette is the cleanest case because every bet resolves on a single spin over a known number of pockets. Take a straight-up bet (one number) on a single-zero European wheel: 37 pockets, and the bet pays 35 to 1.
EV = (+35 × 1/37) + (−1 × 36/37) = (35 − 36)/37 = −1/37 ≈ −2.70%
The double-zero American wheel adds a 00, making 38 pockets while keeping the 35-to-1 payout. The single extra losing pocket is the entire difference:
EV = (+35 × 1/38) + (−1 × 37/38) = (35 − 37)/38 = −2/38 ≈ −5.26%
Every bet on a single-zero wheel carries the same 2.70% edge; every bet on a double-zero wheel the same 5.26%. The payout structure is engineered so the edge is uniform across bet types.III. The craps pass line, derived
The pass line looks complicated but resolves into a single fraction. On the come-out roll you win on 7 or 11 and lose on 2, 3 or 12; any other number becomes the "point," which you must repeat before rolling a 7.
| Come-out result | Probability | Then win the bet with probability… |
|---|---|---|
| 7 or 11 (immediate win) | 8/36 | 1 (already won) |
| 2, 3 or 12 (immediate loss) | 4/36 | 0 |
| Point 4 or 10 | 6/36 | 1/3 (repeat before a 7) |
| Point 5 or 9 | 8/36 | 2/5 |
| Point 6 or 8 | 10/36 | 5/11 |
P(win) = 8/36 + (6/36)(1/3) + (8/36)(2/5) + (10/36)(5/11) = 244/495 ≈ 0.4929
house edge = 1 − 2 × (244/495) = 7/495 ≈ 1.41%
Because the pass line pays even money, the edge is just one minus twice the win probability — a tidy 1.41%. Backing it with free odds (paid at true odds, zero edge) dilutes the combined figure toward 0.4% or lower.IV. Baccarat and blackjack — same definition, computed by enumeration
Not every game collapses to a one-line fraction. Baccarat resolves over multi-card sequences governed by its drawing tableau; blackjack adds player decisions on top. For these the edge is obtained by enumerating (or simulating) every sequence against the fixed rules and applying the very same EV formula. The output is still a single derived number:
- Baccarat Banker — enumerate the eight-deck tableau: EV ≈ −0.0106 → house edge ≈ 1.06%. Full working in the baccarat derivation.
- Blackjack, basic strategy — evaluate the optimal action for every hand against every up-card: house edge ≈ 0.4–0.5% on a favourable six-deck game. The chart itself is derived in the basic-strategy derivation.
Premise. Baccarat: standard eight-deck shoe. Blackjack: six decks, dealer stands on soft 17, double-after-split and surrender allowed, 3:2 naturals, perfect basic strategy. Change the rules and the number moves — see how rule variants move the house edge.
V. The full reference table
| Game — bet | House edge | Premise |
|---|---|---|
| Craps — Pass line + 5× odds | ≈ 0.37% | Combined bet, odds paid at true odds |
| Blackjack — basic strategy | ≈ 0.4–0.5% | 6 decks, S17, DAS, 3:2 |
| Baccarat — Banker | ≈ 1.06% | 8 decks, 5% commission |
| Baccarat — Player | ≈ 1.24% | 8 decks |
| Craps — Pass line (no odds) | ≈ 1.41% | Even-money base bet |
| Roulette — European | ≈ 2.70% | Single zero, any bet |
| Roulette — American | ≈ 5.26% | Double zero, any bet |
| Baccarat — Tie | ≈ 14.4% | 8:1 payout |
Every figure carries a premise. The game name alone never fixes the edge; the rule set does. The complete sorted table lives in the downloadable house-edge reference dossier.
VI. Edge, hold and element of risk — three different numbers
- House edge — theoretical loss per unit bet. Fixed by the rules. This is the number derived above.
- Element of risk — theoretical loss per unit resolved, which differs for bets that can be raised mid-hand (doubling, odds). Useful for comparing games with different average action per decision.
- Hold — the casino's operational take as a share of the money players buy in with. Because winnings get re-bet, hold runs several times the house edge; it is a business metric, not a property of the game.
- One formula fixes every edge: house edge = −EV ÷ stake.
- Roulette (1/37, 2/38) and the craps pass line (7/495) are exact closed-form fractions.
- Baccarat and blackjack use the same definition, computed by enumeration against fixed rules.
- Every figure is only meaningful with its rule premise attached.
- House edge, element of risk and hold are three distinct numbers — don't conflate them.