True odds and the price paid
Every casino bet involves two numbers. One is the true chance of winning, set by the game’s physical or digital mechanics. The other is the payout the casino offers if you win. The house edge exists because the payout is lower than the true odds would justify. If the two matched, the bet would be fair and the casino would have no built-in advantage.
Roulette is a clean place to see this because its rules are simple and its numbers are standard.
Single zero roulette
A European-style wheel has 37 pockets: the numbers 1 to 36 and a single zero. Each spin is independent, and each pocket is equally likely, so the chance of any one number is 1/37.
A straight-up bet on one number pays 35 to 1: you keep your stake and receive 35 times the stake. Consider a stake of 1 unit.
- With probability 1/37 you win 35 units.
- With probability 36/37 you lose 1 unit.
The expected value (the average result per spin over many spins) is:
(1/37) × 35 + (36/37) × (-1) = (35 - 36)/37 = -1/37
That is about -0.027, a loss of roughly 2.70% of each stake on average. This figure is the house edge for this bet.
What would a fair payout be? With 37 equally likely pockets, the true odds against a given number are 36 to 1. A fair game would pay 36 to 1. The casino pays 35 to 1, and that one unit of difference, spread over 37 outcomes, is the whole edge.
Double zero roulette
The American-style wheel adds a second zero: 38 pockets, with the same 35 to 1 payout on a single number.
- With probability 1/38 you win 35 units.
- With probability 37/38 you lose 1 unit.
Expected value: (35 - 37)/38 = -2/38 = -1/19, about -0.0526, or a loss of roughly 5.26% per unit staked.
Nothing about the payout changed. The extra pocket made winning less likely while the payout stayed the same, which roughly doubles the edge. Different wheels with different rules give different numbers, so always check which variant is being described.
The edge is the same on other roulette bets
Take an even-money bet such as red, which pays 1 to 1. On a single zero wheel, 18 pockets are red, 18 are black, and the zero is neither.
- Win with probability 18/37, gaining 1.
- Lose with probability 19/37, losing 1.
Expected value: (18 - 19)/37 = -1/37, again about -2.70%. On a double zero wheel it is (18 - 20)/38 = -2/38, about -5.26%.
So a bet that looks closer to 50-50 loses at the same average rate as a long shot. The size of the edge is the same; what differs is how bumpy the ride is. Some wheels use special rules when the ball lands on zero, which can reduce the edge on even-money bets, so rules matter.
Return to player
The same information can be expressed as a return: the fraction of money staked that comes back on average. For the single zero straight-up bet that is 36/37, about 97.3%. For the double zero wheel it is 36/38, about 94.7%. A house edge of 2.70% and a return of 97.3% are the same fact described from two sides.
What a house edge means in practice
Suppose, purely as an illustration, that someone staked 1 unit on each of 1,000 single zero spins. The total staked is 1,000 units. At an average loss of 1/37 per unit, the expected loss is 1,000/37, about 27 units. On a double zero wheel it would be about 1,000 × 2/38, roughly 52.6 units.
These are averages, not predictions. In any one session the result can be better or much worse, and the swings are usually larger than the edge. That is why short-term results can mislead: a lucky session says little about the long-run advantage. This is the law of large numbers at work, and it is why the casino can rely on the edge across thousands of bets even though any one player may be ahead for a while. The edge only needs time and volume to show up, and it applies to the total staked, not to the original bankroll.
How this extends to other games
The same logic applies everywhere in the casino. For each bet, compare the true chance of winning with the payout:
- Fair price = payout that would give an expected value of zero.
- House edge = the shortfall between that and what is actually paid.
Games with fixed outcomes, like roulette, have an edge you can calculate exactly from the rules. Games that combine chance with decisions, such as blackjack, have an edge that depends on the rules and on how the player acts, so no single number applies unless the rules are fully specified. Do not trust a figure quoted without its rules. And games with long-shot jackpots hide a negative expected value inside rare, large prizes.
Common misunderstandings
“The wheel is due.” Each spin is independent. After a run of reds, the next spin has the same chances as any other. Believing otherwise is the gambler’s fallacy.
“Changing bets changes the edge.” Moving between bets on the same wheel does not make the average loss disappear, because every standard bet carries a negative expected value, and combining negative-expectation bets gives a negative-expectation total.
“A small edge is harmless.” A 2.70% edge sounds small, but it applies to every unit staked, so it compounds with the total amount wagered.
A note on risk
No calculation turns a negative expected value into a profit. These numbers explain why casino games are designed the way they are, not how to get ahead of them. Gambling can be harmful, and outcomes are uncertain even when the average is known. Never bet more than you can afford to lose or borrow to play, and if gambling is causing difficulty, contact a local support service.