--- title: "House edge" url: https://gagamblinghelpline.org/terms/house-edge/ description: "How the house edge is defined and derived: the break-even prize, the single-zero and double-zero wheel figures, and why the edge applies to turnover rather than to." --- # House edge ## Topic chapters - [Probability](https://gagamblinghelpline.org/terms/#C) (10) - [Lotteries and Draws](https://gagamblinghelpline.org/terms/#C) (4) - [Odds and Prices](https://gagamblinghelpline.org/terms/#B) (8) - [House Advantage](https://gagamblinghelpline.org/terms/#A) (10) - [Variance and Streaks](https://gagamblinghelpline.org/terms/#B) (11) - [Games and Rules](https://gagamblinghelpline.org/terms/#A) (7) ## Long-form chapters - [Probability and expected value](https://gagamblinghelpline.org/topics/probability-and-expected-value/) - [How the house edge is calculated](https://gagamblinghelpline.org/topics/how-house-edge-is-calculated/) - [Odds notation and implied probability](https://gagamblinghelpline.org/topics/odds-notation-and-implied-probability/) - [Variance, spread and why streaks are ordinary](https://gagamblinghelpline.org/topics/variance-and-streaks/) - [Independence and the gambler's fallacy](https://gagamblinghelpline.org/topics/independence-and-the-gamblers-fallacy/) - [Return to player and lottery prize funds](https://gagamblinghelpline.org/topics/return-to-player-and-lottery-prize-funds/) - [How games of chance are designed](https://gagamblinghelpline.org/topics/how-games-of-chance-are-designed/) - [A history of lotteries and betting](https://gagamblinghelpline.org/topics/history-of-lotteries-and-betting/) - [The development of gambling regulation](https://gagamblinghelpline.org/topics/development-of-gambling-regulation/) ## Letters carrying entries [A](https://gagamblinghelpline.org/terms/#A) [B](https://gagamblinghelpline.org/terms/#B) [C](https://gagamblinghelpline.org/terms/#C) [D](https://gagamblinghelpline.org/terms/#D) [E](https://gagamblinghelpline.org/terms/#E) [F](https://gagamblinghelpline.org/terms/#F) [G](https://gagamblinghelpline.org/terms/#G) [H](https://gagamblinghelpline.org/terms/#H) [I](https://gagamblinghelpline.org/terms/#I) [J](https://gagamblinghelpline.org/terms/#J) [K](https://gagamblinghelpline.org/terms/#K) [L](https://gagamblinghelpline.org/terms/#L) [M](https://gagamblinghelpline.org/terms/#M) [N](https://gagamblinghelpline.org/terms/#N) [O](https://gagamblinghelpline.org/terms/#O) [P](https://gagamblinghelpline.org/terms/#P) [Q](https://gagamblinghelpline.org/terms/#Q) [R](https://gagamblinghelpline.org/terms/#R) [S](https://gagamblinghelpline.org/terms/#S) [T](https://gagamblinghelpline.org/terms/#T) [U](https://gagamblinghelpline.org/terms/#U) [V](https://gagamblinghelpline.org/terms/#V) [W](https://gagamblinghelpline.org/terms/#W) [Z](https://gagamblinghelpline.org/terms/#Z) Twenty-four of twenty-six letters hold at least one entry. X and Y hold none. ## Figures at a glance - Single number, 37 pockets, paid 35 to 1 **2.70 %** per unit staked - Single number, 38 pockets, paid 35 to 1 **5.26 %** per unit staked - Dice sequence, pass on 7 or 11 **1.41 %** per unit staked - Two-way market priced 1.91 each side **4.50 %** per unit staked Each figure is derived on the chapter that states it. ## Editorial policy Figures are derived on the page that states them, and each is printed with the base it is measured against. [Read the method](https://gagamblinghelpline.org/method/) House Advantage The house edge is the proportion of each amount staked that a game is expected to retain, once every outcome has been weighted by its probability. per unit staked The edge is not a fee added to a game and it is not a share taken from winnings. It is the difference between the prize a game pays for a result and the prize that same result would have to pay for the game to break even. Fix the prize below that break-even figure and the edge exists; it is a property of the rules, present before any wheel turns and unaffected by anything that happens afterwards. ### The break-even prize A result that occurs on one trial in *n* must pay *n* − 1 units of profit per unit staked for the bet to break even: the single winning trial has to replace the stake lost on each of the other *n* − 1 trials. On a wheel of thirty-seven pockets a single number occurs once in thirty-seven, so the break-even prize is 36 to 1. The prize actually paid is 35 to 1. The entire edge is that one unit of shortfall, spread across thirty-seven trials. ``` Single number, 37 pockets, paid 35 to 1 win 1/37 x (+35) = +0.945946 lose 36/37 x ( -1) = -0.972973 ----------- expected value = -0.027027 per unit staked as a fraction: (35 - 36) / 37 = -1/37 = -2.70 % ``` ### The same edge on a different bet An even-money bet on the same wheel covers eighteen of the thirty-seven pockets and pays 1 to 1. The arithmetic differs but the answer does not, because the single pocket lying outside both eighteen-pocket groups is the whole of the shortfall in either case. ``` Even-money group, 37 pockets, paid 1 to 1 win 18/37 x (+1) = +0.486486 lose 19/37 x (-1) = -0.513514 ----------- expected value = -0.027027 per unit staked = -2.70 % ``` ### Adding a second unmatched pocket A wheel of thirty-eight pockets that still pays 35 to 1 on a single number has two pockets outside the paying structure rather than one, and the edge very nearly doubles. ``` Single number, 38 pockets, paid 35 to 1 win 1/38 x (+35) = +0.921053 lose 37/38 x ( -1) = -0.973684 ----------- expected value = -0.052632 per unit staked as a fraction: (35 - 37) / 38 = -2/38 = -1/19 = -5.26 % ``` Two wheels with identical prizes and nearly identical appearance therefore differ by a factor of just under two in what they are expected to retain, and the difference is visible only in the pocket count. This is why an edge cannot be judged from prizes alone: the prize is one half of a comparison, and the count of ways to lose is the other. ### What the figure applies to The edge is stated per unit staked, which means it applies to turnover and not to the amount a player began with. A figure of 2.70 % describes what happens to each unit as it is staked, and a unit that is won back and staked again is counted again. The consequence is arithmetical: expected loss grows in proportion to total amounts staked, without limit, as the count of trials rises. ## Related entries [Return to player](https://gagamblinghelpline.org/terms/#R) The proportion of total amounts staked a game returns as prizes over the long run; its complement is the house edge. per unit staked HOUSE ADVANTAGE [True odds](https://gagamblinghelpline.org/terms/#T) The odds corresponding exactly to a result's probability, against which the odds actually paid are compared. HOUSE ADVANTAGE [Negative expectation](https://gagamblinghelpline.org/terms/#N) A bet whose probability-weighted average outcome is a loss, which is the condition of every commercially offered game of chance. HOUSE ADVANTAGE [Expected value](https://gagamblinghelpline.org/terms/expected-value/) The probability-weighted average of every outcome of a bet; on a 37-pocket wheel it is minus one part in thirty-seven. per unit staked PROBABILITY [Zero](https://gagamblinghelpline.org/terms/#Z) The pocket or pockets on a wheel belonging to neither even-money group, and the entire source of the wheel's edge. HOUSE ADVANTAGE [Hold](https://gagamblinghelpline.org/terms/#H) The proportion of the amount staked that a game or table retains once play has finished. per unit staked HOUSE ADVANTAGE