House edge and expected value

The calculation

Expected value is the average result of a wager per unit staked, worked out in advance rather than observed. The method is mechanical. List every outcome, write down what the wager returns in each case, multiply each return by the probability of that outcome, and add the products together.

Take a wager on a single number of a roulette wheel of the single-zero design, which carries thirty-seven pockets numbered zero to thirty-six. The wager wins if the ball settles in one specified pocket, which happens with probability one in thirty-seven, and it pays thirty-five to one. Win and the player is thirty-five units up; lose and the player is one unit down, which happens with probability thirty-six in thirty-seven.

The expected value is therefore thirty-five multiplied by one thirty-seventh, plus minus one multiplied by thirty-six thirty-sevenths. That is 35/37 minus 36/37, which is minus one thirty-seventh, or minus 0.0270. Expressed as a percentage the wager returns minus 2.70 per cent of the stake on average. The house edge for that game is 2.70 per cent.

Why the number does not change with the bet chosen

The striking feature of that wheel is that almost every wager available on it produces the same answer. A wager on red covers eighteen of the thirty-seven pockets and pays one to one. Its expected value is one multiplied by eighteen thirty-sevenths, minus one multiplied by nineteen thirty-sevenths, which is minus one thirty-seventh again. A wager on a column covers twelve pockets and pays two to one: two times twelve thirty-sevenths minus twenty-five thirty-sevenths, once more minus one thirty-seventh.

This is not a coincidence. The payouts were derived from a wheel of thirty-six numbers and then applied to a wheel with one extra pocket. Every payout is one unit short of the figure that would make the wager break even, and one unit short of a thirty-seven pocket wheel is exactly one thirty-seventh. The edge is built into the mismatch between the paytable and the sample space, so changing which wager is placed does not escape it.

Add a second zero pocket, as the double-zero design does, and the same reasoning gives a different answer. The wheel now has thirty-eight pockets, a single number still pays thirty-five to one, and the expected value becomes thirty-five times one thirty-eighth minus thirty-seven thirty-eighths, which is minus two thirty-eighths, or minus 5.26 per cent. One extra pocket with no corresponding change to the paytable very nearly doubles the edge.

Edge, hold and the volume of play

House edge is defined per unit staked, which is why it does not describe how much of a player's money a session consumes. That quantity depends on how many times the money is staked. A player who brings one hundred units and stakes them once faces an average loss of 2.70 units on the wheel described above. A player who stakes one unit at a time for four hundred spins has staked four hundred units in total and faces an average loss of roughly 10.8 units, even though the money on the table at any moment was never more than one.

The distinction between the two figures is the difference between edge and hold. Edge is a property of the rules. Hold is what the rules produce when combined with a rate of play, and it is the second of those numbers that determines what a session actually costs. Any design decision that increases the number of wagers made per hour increases the total retained without altering the edge at all.

The one result that follows

If the expected value of every available wager is negative, then the expected value of any combination of those wagers is also negative, because expectation adds. No staking pattern, no sequence, no rule about when to increase or decrease the stake changes this, since each of those is only a way of choosing which negative-expectation wagers to make and in what order.

Systems that appear to defeat the arithmetic always do so by hiding the loss somewhere else. Doubling after every loss, for example, converts a small chance of a large loss into a large chance of a small gain, which feels like an improvement and is not one: the two adjustments cancel exactly, and the table limit that caps the doubling makes the trade slightly worse than exact. The expected value of the sequence is the sum of the expected values of its parts, and each part is negative.