Expected value (also called mathematical expectation) is the single most important characteristic of a random event in casino games.
It represents the average value of the random variable – the profit that a casino (or a player) will earn over a long enough horizon of play. In practical terms, it answers the question: if this game were played indefinitely, what would the average result per bet converge to?
When expressed as a percentage of a single wager, this metric is known as House Edge – the share of each bet that the player will, on average, lose over the course of extended play.
A negative expected result for the player is, by definition, a positive expected result of equal magnitude for the casino – and vice versa, in the rare cases where the player holds an edge (Player Edge).
House Edge is the inverse of Player Edge. It is the metric most commonly used to characterize the casino’s edge – an indicator of a game’s profitability. It represents the percentage the casino retains from every bet placed by the player over a long horizon.
Expected Value is an absolute measure – the dollar amount the player (or casino) can expect to win or lose on a bet of a specific size over a long period. It depends on both the type and the size of the bet. House Edge, by contrast, is a relative measure – a percentage that remains constant regardless of bet size.
On Roulette, two bets of different sizes on different positions will produce different expected values but an identical House Edge.
Both Expected Value and House Edge are long-run metrics. Actual results approach their theoretical values only after a large number of repetitions. The more repetitions, the closer the actual results converge to the expected average.
Calculating Expected Value and House Edge
Where n is the total number of possible outcomes. Each of the i = 1…n possible outcomes has a probability Pi and a result Xi.
The probabilities of all possible outcomes sum to one – meaning every possible result is accounted for, and one of them is certain to occur. The result of each outcome is the net win (excluding the original wager), which is typically paid in proportion to the bet placed – or a loss equal to the bet itself.
Common Mistakes in Interpreting House Edge
House Edge is a critical metric, but in practice it is frequently misinterpreted and misapplied.
You cannot compare games solely by House Edge and conclude which is more profitable. A game with a lower House Edge can generate more revenue for the casino than one with a higher edge – because revenue depends on volume, speed of play, average bet size, and other factors that House Edge alone does not capture.
House Edge is calculated against the initial bet only. In games with options – such as Blackjack, draw poker, or any game where the player can make additional wagers during a round – those supplementary bets are not included in the House Edge calculation. On Blackjack, for instance, House Edge represents the casino’s hold as a percentage of the initial wager in each hand. When a player uses available options (Split, Double, Surrender) only in favorable situations, the casino’s effective edge – measured against the average total amount wagered per hand – is considerably lower. Dividing House Edge by the average bet per hand produces a more flexible and analytically useful metric known as Element of Risk.
On Roulette, Baccarat, and other games where no in-round options exist and the initial bet remains unchanged, House Edge and Element of Risk are identical.
In games with unstable expected value – games with options like Blackjack and Poker – the average expected result and House Edge depend on the quality of the player’s decisions, their skill level, and their ability to make optimal choices. For these games, House Edge is calculated assuming perfect play -flawless decision-making on every option the rules allow.
House Edge only works over the long run, across a very large number of repetitions. Over a short playing session, it is not a useful metric for analyzing actual results. Short-term outcomes are governed by variance, not by expected value alone.
