Every casino game pays its winnings slightly below the level that would match the real chances. The difference, expressed as a share of the stake, is called the house edge. Its complement, the share of stakes that goes back to players in the long run, is called RTP ("return to player"). Both are calculated from the rules of the game with the same tool: the average result of a bet.
The average result of a bet
The average result, called the expected value in mathematics, is obtained by multiplying each possible result of the bet by its probability and adding up the products:
expected value = Σ (probability of the result × amount won or lost in that case)
In European roulette the wheel has 37 pockets (0–36). A bet on a single number (straight up) pays 35 to 1: it wins 35 units with probability 1/37 and loses the unit staked in the other 36 cases out of 37. Per unit staked:
expected value = (1/37) × 35 + (36/37) × (−1)
The result is an average loss of 1/37 of the stake, that is, 2.70%. Per 100 units staked, the average result is −2.70 units. A fair bet, with no edge for either side, would pay 36 to 1: the 36 losing pockets against the winning one. The house pays 35 to 1, and the difference is its edge.
The same calculation for a bet on red, which pays 1 to 1 and covers 18 pockets, gives the same average loss: (18/37) × 1 + (19/37) × (−1). The zero is what creates the edge, on every bet on the table. The full table of bets is on the Roulette page.
The house edge
The house edge is the player's average loss relative to the initial stake, with the sign reversed: the share of each stake that the house keeps, on average, over a very large number of bets. It is a property of the bet, not of the player or of the session. It does not say how much someone loses in an evening, but how much they end up losing, on average, per unit staked, once the number of bets becomes very large.
In some games the edge also depends on the player's decisions. The conditions of the calculation are then stated: which rules, and which way of making decisions.
RTP
RTP is the share of stakes returned to players in the long run: RTP = 1 − house edge. In European roulette, the RTP is 97.30%. The term is used above all for gaming machines, where the probabilities cannot be seen: the reels and the paytable are set by the maker, and each game's RTP is calculated from them and stated for each game.
The Slot machines page describes an imaginary machine, built for the site, whose reels and payouts are all visible. Its RTP is 90.69%, so the house edge is 9.31%. The machine does not represent any real machine; the example only shows how RTP follows from probabilities and payouts.
Examples from several games
The house edge differs from one game to another and often from one bet to another within the same game:
- Baccarat. The banker bet has a house edge of 1.06%, after the commission taken from winnings on that bet. The player bet has an edge of 1.24%, and the tie bet one of 14.36%. Details are on the Baccarat page.
- Blackjack. Here the edge depends on the player's decisions and on the rules. With basic strategy, meaning the decision with the best average result in every situation, and with the rules described on the Blackjack page, the house edge is 0.57%. If the dealer also hits soft 17, it rises to 0.79%. A player who played the same way as the dealer would face an edge of 5.67%. Decisions change the average loss, but the house edge remains in every case.
- Craps. The pass line bet has a house edge of 1.41%. The one-roll bet on 7 has an edge of 16.67%. The full list is on the Craps page.
- Sports betting. The edge is not written in the rules but built into the odds. In the example with invented odds on the Betting odds page, the organiser keeps on average 4.59% of every stake.
- European roulette. 2.70% on every bet on the table, as shown above.
The differences between bets show that the house edge is calculated for each bet separately, not only for a game as a whole.
Why RTP is an average, not a promise
RTP describes the average over a very large number of bets. It does not describe the result of a session. An RTP of 97.30% does not mean that a player who stakes 100 will get back that share of it. In a short session, the result can be far above the average or far below it: on a single-number bet, a spin ends either with the full win or with the loss of the stake, never with the average. How far results can stray from the average is measured by variance, explained in Variance and the law of large numbers.
Two points matter for reading these figures correctly:
- The average applies to the total amount staked, not to the starting money. The same money, staked again after every spin or hand, is exposed to the house edge each time. The average loss of a session is the house edge multiplied by everything staked in it, and that amount can be many times larger than the sum the player started with.
- The average is over everyone, not per person. A machine's RTP refers to all the spins played on it, by all players. Some finish ahead, most finish behind; their average, over a very large number of spins, approaches the RTP.
For the house, which takes a very large number of bets, the total result ends up very close to the calculated edge. For a player, the edge is the direction the average moves towards as bets add up. No way of choosing stakes changes that direction, as the article on Betting systems shows.
See also
- Probability basics
- House edge, RTP and variance in the glossary