What is the house edge?
The house edge (HE) is the average percentage of each bet the casino expects to keep over the long run. For a game with 49% win probability and even-money payouts, HE = 1 โ 2ร0.49 = 2%. It is the negative expected value expressed as a percentage of the wager.
- European roulette: HE โ 2.70% (one zero, 37 pockets)
- American roulette: HE โ 5.26% (two zeros, 38 pockets)
- Blackjack (basic strategy): HE โ 0.5%
- Baccarat banker bet: HE โ 1.06%
- Video poker (9/6 Jacks or Better): HE โ 0.46%
Expected value and return-to-player
Return-to-player (RTP) = 100% โ HE. A slot with 96% RTP returns $0.96 per $1 wagered on average. The expected loss per session is HE ร bet_size ร number_of_bets. This is a long-run average โ individual sessions vary widely due to variance.
- EV per bet = p_win ร payout โ p_lose ร 1
- A negative EV game cannot be beaten in the long run by strategy alone
- Comps and bonuses effectively reduce the house edge for some players
Variance and standard deviation
Games with the same HE can feel very different because of variance. High-variance games (slots, parlays) have large swings; low-variance games (baccarat) show smoother results. The standard deviation of a session is ฯ = bet_size ร โn ร ฯ_per_bet.
- Higher variance = more chances of a big win, but also bigger potential loss
- Short sessions are dominated by variance; the HE dominates over thousands of bets
- The Central Limit Theorem explains why the average converges to the HE over time