What Is a House Edge and How Is It Worked Out?
The house edge is the average amount a bet loses per pound staked over a very long run. It is the expected value of the bet with the sign turned round: add up every outcome’s probability multiplied by what it pays or costs.
- List the outcomes. Every way the bet can end: win, lose, push, or several winning amounts.
- Find each probability. Count the ways each outcome can happen and divide by all possible ways.
- Multiply by the result. A win of 35 units, a loss of 1 unit, a push of 0.
- Add them up. The total is the expected value; a negative total is the house edge.
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Written as a formula, the expected value of a bet is the sum of probability × result across all outcomes, measured in units of the stake. If a £1 bet has an expected value of minus 2.7p, its house edge is 2.7%. Nothing about a single bet is guaranteed by that number; it describes what happens to the total when a bet is repeated many thousands of times.
We count outcomes exactly wherever the game allows it. Cards and dice have a fixed number of possible results, so the probabilities are fractions that can be checked by hand. For slot machines this is not possible from the outside, because each game’s maths is set by its maker; there we use the return to player (RTP) the game publishes, and the house edge is simply 100% minus the RTP.
Roulette: Why the Edge Is 1 in 37
A single-zero wheel has 37 pockets but pays bets as if it had 36. A straight-up bet wins 35 units with probability 1/37 and loses 1 unit with probability 36/37, so its expected value is (35 − 36) ÷ 37 = −1/37, a house edge of 2.70%.
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The same edge applies to almost every bet on a single-zero wheel, because every payout is set to the same fair rate on 36 numbers. A red or black bet wins with 18 pockets out of 37 and loses with 19: (18 − 19) ÷ 37 = −1/37 again. On a double-zero wheel there are 38 pockets, and the edge on most bets becomes 2 ÷ 38, or 5.26%.
La partage changes only the even-money bets. When the ball lands on zero, half of an even-money stake is returned, so the zero costs half a stake instead of a whole one and the edge on those bets halves to 1/74, or 1.35%. Our European vs American roulette page sets the wheels side by side.
Poker Hands: Counting 2,598,960 Combinations
A 52-card deck can be dealt into 2,598,960 different five-card hands, the number of ways to choose 5 cards from 52. The probability of any hand type is the number of ways to make it divided by that total.
| Hand | Counting method | Hands | Probability |
|---|---|---|---|
| Royal flush | One per suit | 4 | 1 in 649,740 |
| Four of a kind | 13 ranks × 48 fifth cards | 624 | 0.024% |
| Full house | 13 × 4 (three of a rank) × 12 × 6 (pair of another) | 3,744 | 0.144% |
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The total itself comes from the combination formula: 52 × 51 × 50 × 49 × 48 divided by 5 × 4 × 3 × 2 × 1, because the order in which the cards arrive does not matter. Each hand type is counted the same way, by choosing its ranks and then its suits, and the ten counts add back up to 2,598,960 as a check. The full table is on our poker hand rankings page.
A hand ranks higher the fewer ways there are to make it. That is why a flush beats a straight: there are 5,108 flushes (leaving out straight flushes) against 10,200 straights.
Baccarat: Probabilities From Every Card Combination
Baccarat’s drawing rules are fixed, so every outcome follows from the cards. Working through every combination of an eight-deck shoe gives Banker 45.86%, Player 44.62% and Tie 9.52%.2 (source)
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From those three figures the edges follow by the method above, with ties treated as a push for Banker and Player bets. Banker pays 0.95 after the 5% commission: 0.4586 × 0.95 − 0.4462 = −0.0105, a house edge of 1.06% once the exact probabilities are used. Player pays even money: 0.4462 − 0.4586 = −0.0124, or 1.24%. A Tie bet paying 8 to 1 returns 0.0952 × 8 − 0.9048 = −0.1432, a house edge of 14.36% once the exact probabilities are used.
Banker wins more often because it acts second under the third-card rule, which is why the casino charges a commission on it. Our baccarat rules page shows the full third-card table.
Craps: The Pass Line in Fractions of 495
Two dice give 36 equally likely results. Adding the chance of winning on the come-out roll to the chance of making each point gives a pass line win probability of 244/495, so the bet loses 7/495 of each stake, a house edge of 1.41%.
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On the come-out roll, 7 or 11 wins (8 ways in 36) and 2, 3 or 12 loses (4 ways). Any other total becomes the point, which must be rolled again before a 7. A point of 4 is made 3 times for every 6 sevens, so it is won with probability 3/9; points of 5 and 9 with 4/10; points of 6 and 8 with 5/11. Weighting each point by how often it is set and adding the come-out wins gives 244/495, or 49.29%.
The free odds bet that can be laid behind the pass line pays exactly those true odds, 2 to 1 on a 4 or 10, 3 to 2 on a 5 or 9 and 6 to 5 on a 6 or 8, which is why it carries no house edge of its own.
Blackjack: Exact Dealer Odds and Published Rule Effects
For the dealer’s final hand we calculate every possible draw from each up-card, applying the rule that the dealer draws to 16 and stands on all 17s. A full blackjack house edge depends on thousands of player decisions, so for rule effects we use published figures.1 (source)
Swipe the table
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | A |
|---|---|---|---|---|---|---|---|---|---|
| 35.4% | 37.4% | 39.4% | 41.6% | 42.3% | 26.2% | 24.5% | 22.8% | 21.2% | 11.5% |
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The calculation starts from the up-card and branches on every possible next card. A ten-value card has a weight of 4/13, because four ranks count as ten, and every other rank has a weight of 1/13. Each branch either ends, when the dealer reaches 17 or more or goes over 21, or branches again. An Ace counts as 11 until that would take the hand over 21, then as 1. Adding the weighted branches gives the probability of each final total, and each up-card’s results add up to exactly 100%.
Simpler blackjack figures are calculated directly. Insurance pays 2 to 1 and wins when the dealer’s next card is one of the four ten-value ranks: 2 × 4/13 − 9/13 = −1/13, a house edge of about 7.7%. A natural blackjack from a six-deck shoe needs one of 24 Aces and one of 96 ten-value cards in either order: 2 × 24/312 × 96/311 = 4.75%.
The effect of each rule, such as a 6:5 payout or the dealer hitting soft 17, comes from published full-strategy analyses, which we cite on each page. These effects are measured against a stated base game and add up only roughly, because rules interact.
What Assumptions Sit Behind the Figures?
Every figure assumes the rule set stated on its page, correct play where the game involves decisions, and a very long run of bets. Change the rules and the figure changes; play badly and the cost rises.
Before a figure goes on a page we check it a second way. A probability worked out with a formula is compared with a full count of every combination, or with a published figure for the same rule set where one exists. When two methods disagree and we cannot find the reason, the figure does not appear on the site until we can. That is why some pages quote a published estimate rather than our own number: we would rather cite a checked figure than print one we cannot yet stand behind.
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Where a calculation treats the shoe as infinite, we say so. With six or eight decks the difference from a real shoe is small, but it is not zero, and published figures for a specific number of decks take priority when we quote a precise house edge. Where we round, we round to two decimal places for edges and one for probabilities.
A house edge is a long-run average, not a forecast. In a single evening a player can win or lose far more than the edge suggests, in either direction. That is the nature of chance. If you would like to check one of our calculations, or think one is wrong, write to [email protected]; corrections are handled as set out in our editorial policy.
Our Odds Calculations: Frequently Asked Questions
Short answers on how the figures are worked out, why they can differ between sites and what they do and do not tell you.
Why do your figures sometimes differ from other sites?
Do you use simulations?
What does an infinite-deck calculation mean?
Is the house edge the same as my chance of losing?
Sources
Documents behind this page, grouped by publisher type and dated.