Mathematics behind
the games of chance.

Learn about payout ratio and variance

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Principles described in this article can be applied to almost any game of chance in which a player plays against the house. In other words, all games where a casino collects bets and pays out potential winnings. The most famous of such games are: slot machines, roulette, blackjack and baccarat. Most of you probably know that these games are somehow set to give casinos a certain advantage in the long-run. Otherwise, the owners of the casino would be losing money by running those games.

The advantage of the casino is for every game determined by the rules of the game and by the payout rules in the case of winning. If these rules are applied over many, many rounds (sometimes even hundreds of thousands of times), casino owners can be quite certain that the amount of bets collected will exceed the amount of wins payed out. The statistics work in favour of the casino in the long-run.

Despite that long-term statistics work against you, it’s quite common to come to a casino, play, win, and leave with your winnings. The main reason is that your visit to a casino consists only of several tens of game rounds. In this case, the statistics don't have enough time to converge (show up). The outcome of your casino visit is then more determined by chance (or by your luck, if you want). It’s exactly this chance which helps lucky players win and beat the statistical advantage of the casino.

To increase the chances of beating the statistics, it’s very important to know two basic characteristics of each game: payout ratio and variance.

Roulette RTP

In European Roulette, the probability of winning when betting on the black colour is 18 black numbers divided by 37 numbers total (don’t forget the zero). The payout is 2 times the bet.

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The payout ratio of roulette is then:

2 * 18 / 37 = 0.973 = 97.3%

Games of chance and payout ratio (RTP)

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“The higher the size of your bets, the higher (usually) the RTP of your betting system. We assume that all the other rules of the betting system remain the same.”

  • Payout percentage of games of chance applies only for a single round. Your real payout percentage will be lower due to re-betting previous winnings.
  • You statistically lose on every bet. A fewer bets means a smaller statistical profit for the casino.
  • The RTP of your betting system depends on the RTP of the game, variance of the game, size of your bets, and rules when to leave and when to keep playing.
  1. Payout percentage of games of chance applies only for a single round. Your real payout percentage will be lower due to re-betting previous winnings.
  2. You statistically lose on every bet. A fewer bets means a smaller statistical profit for the casino.
  3. The RTP of your betting system depends on the RTP of the game, variance of the game, size of your bets, and rules when to leave and when to keep playing.

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