Foundations

House edge, and where the percentage actually comes from

The edge is not a fee, a commission or a policy. It is the arithmetic gap between what a wager pays and what it should pay, and it can be worked out from the rules alone.

Fanfeud desk note. Figures on this page are worked out from the rules stated beside them.

Every wager in a game of chance has two sets of odds attached to it. The first is the true odds: the real ratio of losing cases to winning cases, counted from the game's own sample space. The second is the payout odds: what the rules of the game undertake to pay a winning wager. If those two numbers were the same, the game would be fair in the technical sense, meaning that over any number of rounds the average result per unit staked would be zero. Games of chance are built so that they are not the same.

The distance between them is the house edge. It is stated as a percentage of everything staked, and it is derived, not decided at the table. Take a single number on a wheel with thirty-seven pockets. There is one winning pocket and thirty-six losing ones, so the true odds against are thirty-six to one. The payout is thirty-five to one. That single missing unit is the whole mechanism.

Expected value is the whole of it

The formal name for that calculation is expected value. You list every outcome the wager can have, multiply each result by its probability, and add them up. The number that falls out is the average result per unit staked, and it does not describe any single round. No one loses 2.70 per cent of a unit on a spin. They win thirty-five or they lose one. The percentage is what those two possibilities average to, and averages only become visible over volume.

Expected value is also the reason that a game's edge cannot be argued with by changing bet size, changing bet type or changing the moment at which you play. Doubling a stake doubles both the win and the loss, so the ratio is untouched. Waiting for a run to end does not help either, because a wheel and a pair of dice have no record of what they did last.

Edge is a property of the rules, not of the venue

Because the edge is derived from the pocket count and the payout schedule, it belongs to the wager rather than to the place the wager is made. Two rooms offering identical rules on the same wager have identical edges, and no amount of decoration changes the arithmetic. What differs between one game and another is how the rules are written: how many pockets, how many decks, how the paytable is weighted, and whether a supplementary wager is paid at true odds.

Edge, hold and return to player

Three figures are often used interchangeably and should not be. The edge is the average retained share of each unit staked. Return to player is simply one minus the edge, written from the other direction, so a 2.70 per cent edge is a 97.30 per cent return. Hold is a different animal entirely: it is the proportion of the money a player brings that stays behind at the end of a session. Hold is usually far larger than the edge, because the same money is staked repeatedly. A unit that is wagered ten times is exposed to the edge ten times.

Where a wager can have no edge at all

One familiar wager has an edge of exactly zero: the free odds bet placed behind an established point at a dice table, which is paid at true odds by design. It is worth knowing about not because it changes anything over a session but because it demonstrates the definition. A wager paid at true odds has an expected value of zero, and it is the only kind of wager in this family of games of which that is true. Every other line on the board pays less than the true odds, which is precisely why the board exists.

The rest of this desk applies the same procedure to specific games. Nothing that follows requires a new idea. It requires only the sample space of the game, the payout the rules state, and the subtraction between them.