The Formula Behind Every Casino Bet
Every house edge figure in gambling — from the 1.06% on baccarat banker to the 40% on keno — comes from the same calculation. It is called expected value, and it represents what a bet returns to the player on average per dollar wagered. A house edge of 5% means the expected value is negative 5 cents per dollar. The casino keeps 5 cents and returns 95 cents on average over many repetitions.
The formula is straightforward:
The Core Formula
Expected Value (EV) = P(win) × Payout − P(lose) × 1
Where P(win) is the probability of winning, Payout is the amount won per dollar bet, and P(lose) is the probability of losing. If EV is negative, the house has an edge equal to the absolute value of EV expressed as a percentage.
A Simple Worked Example — The Coin Flip
Start with a fair coin flip. Heads you win $1. Tails you lose $1. P(win) = 0.5. P(lose) = 0.5. Payout = 1.
EV = 0.5 × 1 − 0.5 × 1 = 0.5 − 0.5 = 0.00
House edge = 0.00%. A perfectly fair bet. Neither side has an advantage. This is why a fair coin flip has no house edge — the probability and payout are perfectly balanced.
Now change the rules slightly. Same coin flip — but if it lands on its edge (which never happens in theory but imagine it could, 1% of the time), the house wins. Heads you win, tails you lose, edge the house wins.
P(win) = 0.495. P(lose) = 0.505. Payout = 1.
EV = 0.495 × 1 − 0.505 × 1 = −0.01 = negative 1%
House edge = 1.00%. The casino keeps 1 cent per dollar wagered on average. This is exactly how casino tie rules work — converting some outcomes from player wins to house wins without changing the payout.
How Tie Rules Create House Edge
Most casino card game house edges are generated by tie rules — situations where both sides would otherwise be even but the casino wins instead of the round being a push. Baccarat does this on the main bets via the 5% commission. Beat the Dealer Configuration B does it by having ties on low cards (2-9) go to the house while ties on high cards push. The mathematical effect is the same — some outcomes that would be neutral become house wins.
In a pure high-card comparison with no tie rule, the player and dealer each have exactly a 50% chance of holding the higher card. The house edge is zero. Add a rule that says tied cards lose, and the house edge equals the probability of a tie. This is exactly how the Beat the Dealer Prediction Bet works — tied player cards occur 7.40% of the time, producing a 7.40% house edge.
Combinatorial Enumeration — How Exact Figures Are Calculated
For card games the most accurate method of calculating house edge is exact combinatorial enumeration — counting every possible outcome and its probability explicitly. In a 52-card single deck there are C(52,2) = 1,326 possible two-card hands. In a 312-card six-deck shoe there are C(312,2) = 48,516 possible two-card combinations.
To calculate the house edge on a two-card side bet — like the Beat the Dealer Suited bet — enumerate all 48,516 possible player hands, count how many qualify as suited, multiply by the payout, subtract the non-qualifying hands and divide by total combinations:
| Step | Calculation | Result |
|---|---|---|
Total combinations | C(312,2) | 48,516 |
Suited combinations | 4 suits × C(78,2) | 12,012 |
Non-suited combinations | 48,516 − 12,012 | 36,504 |
P(suited) | 12,012 ÷ 48,516 | 24.759% |
P(non-suited) | 36,504 ÷ 48,516 | 75.241% |
Expected value | 0.24759 × 3 − 0.75241 × 1 | −0.009646 |
House edge | 0.009646 × 100 | 0.9646% |
This is how every figure on this site is calculated — by counting every possible outcome explicitly rather than estimating or borrowing from other sources.
Why Simulation Is Less Accurate
Some house edge figures published elsewhere are derived from simulation — running millions of computer-generated hands and calculating the observed win rate. Simulation approaches the true probability but never reaches it exactly. A simulation of ten million hands may produce a house edge of 0.961% on the Suited bet. Exact enumeration produces 0.9646%. For simple bets the difference is minor. For complex games with multiple decision points the accumulated error in simulation can be meaningful. Where exact enumeration is feasible — as it is for card games with known deck compositions — it always produces more reliable figures.
Return to Player — The Same Number Differently Expressed
Return to Player (RTP) is simply one minus the house edge, expressed as a percentage. A house edge of 1.06% means an RTP of 98.94%. A house edge of 5.26% means an RTP of 94.74%. RTP is commonly used for slot machines and electronic games where the house edge figure is less commonly published.
RTP does not mean you will receive that percentage back on any given session. It is a long-run mathematical expectation. A player who makes 100 bets on a game with 5% house edge expects to lose approximately $5 per $100 wagered — but may win or lose substantially more on any particular visit.
Why No Strategy Overcomes Negative Expected Value
Betting systems — martingale, Fibonacci, D'Alembert and others — claim to overcome the house edge by adjusting bet sizes based on previous results. They cannot. Each hand in a card game or spin in roulette is an independent event. The cards have no memory. The roulette wheel has no memory. A run of losses does not make a win more likely on the next hand. The house edge applies to each individual bet — and the sum of individually negative expected value bets is always a negative expected value total regardless of how the bet sizes are arranged.
No betting system changes the house edge. Systems that increase bets after losses increase the variance of outcomes — you can win more in good sessions and lose more in bad ones — but the expected value per dollar wagered is identical to flat betting. The house edge is a property of the game, not of bet sizing.
The Bottom Line
House edge is probability and payout arithmetic. Once you understand the formula, you can evaluate any bet on any game yourself. High payout and low probability — check whether the probability justifies the payout. Simple even-money comparison — check whether tie rules are eating into the symmetry. Side bet with a complex pay table — add up the weighted outcomes and see where the expected value falls. The casinos know these numbers. This site publishes them. Now you have what you need to decide where your chips belong.