House edge and expected value

Expected value is the average result of a wager, computed by multiplying each possible outcome by its probability and adding the products. It is not a prediction of any single result; no individual play ever returns the expected value in a game whose outcomes are whole units. It is the figure that the average of many results converges towards, and it is the only number that describes a game's design rather than a player's experience of it.

Consider a one-unit stake on a single number of a thirty-seven-pocket wheel paying thirty-five to one. The win occurs with probability 1/37 and returns a profit of 35; the loss occurs with probability 36/37 and costs 1. The expected value is (1/37 x 35) + (36/37 x -1), which is 35 minus 36 over 37, or -1/37 per unit staked: approximately -2.70 per cent. Every other bet available on that same wheel layout, priced against the same thirty-seven pockets, returns exactly the same figure, which is why the layout looks varied while the mathematics behind it does not vary at all.

House edge is expected value expressed as a positive cost, and it always arises the same way: the game pays slightly less than the true odds of the event it is paying on. In the wheel example the true odds against a single number are thirty-six to one, and the game pays thirty-five to one. That single missing unit is the whole of the margin. Remove it—pay the true thirty-six—and the expected value becomes exactly zero and the game stops being a business.

Because the margin is created by the paytable rather than by any event during play, it does not depend on how the player behaves within the rules. Staking more, staking less, spreading a stake across several outcomes or concentrating it on one all leave the per-unit expectation unchanged. Systems that alter the size or the timing of stakes redistribute the shape of the results—making small wins more frequent and losses larger, or the reverse—without touching the multiplier that generates them.

Three figures are used to describe the same design from different angles, and they are frequently confused. House edge is the expected cost per unit staked on a single decision. Return to player is its complement, the proportion of each unit staked that is returned as winnings on average, so an edge of 2.70 per cent corresponds to a return of 97.30 per cent. Hold is an accounting figure: the proportion of the money brought to a game that is not carried away at the end of a session.

Hold is always larger than edge, often several times larger, and the reason is recycling. A unit that is won is frequently staked again, and the edge applies to it a second time, a third, a tenth. A game with a two per cent edge that turns its stake over ten times has taken far more than two per cent of the money that entered it. Quoting a low edge is therefore an accurate but incomplete description of what a session costs; the number of decisions taken matters as much as the margin on each.

Any paytable can be read as a sum. List each winning combination, compute its probability from the mechanism, multiply by what that combination pays, add the products, and subtract one for the stake. The result is the expected value, and the exercise usually takes a few minutes. What it reveals is which lines of the table carry the game's return and which are decoration: in most designs a small number of frequent, low-multiple outcomes supplies almost all of the return, while the large headline payouts contribute very little to the average because their probabilities are so small.

This is why a game can advertise a spectacular top prize and still return less than a plain one. The top line is a marketing surface with almost no weight in the sum; the frequent lines, which nobody quotes, decide the arithmetic. Reading a paytable as an equation rather than as a menu is the most direct way to see the difference.