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Math Theory

It is commonly believed that the main product in the casino is adrenaline. Often we hear that the casino offers to pull the “lucky ticket”, many less often say that the casino sells the service. In fact, the main product of the casino – is the excitement of the opportunity to win. In this article we will consider the basic principles on which the work of gambling houses, the rationale for the profit of the institution, and what role in its activities plays “lady luck.


And we will begin the review by examining the basic mathematical laws on which the gambling is based. How are mathematics and casinos connected? After all, all casino games were invented and developed by mathematicians. Is it possible to use their own weapons to gain an advantage in a gambling house?

Mathematics of casino games

Consider the processes that occur in gambling, in terms of probability theory, and try to determine whether the casino games are subject to mathematics.

Throwing a coin, you can say that any of its sides can fall out with the same probability. There are only two possibilities – the eagle or the tails will fall out. The probability that the flip of the coin will fall out of the lattice is equal ? (50%), i.e. we can expect the tails to fall out in half the cases. Often the word chance is used to describe the probability. The chance of the coin flipping upwards will fall with the tails is 50%.

Probability shows how often the result we expect can be achieved, and can be represented as a ratio of expected outcomes to the total number of all possible outcomes over a sufficiently long period of time with a large number of repetitions.

Mathematical expectation when playing roulette

Let’s calculate a mathematical expectation when playing roulette (the American version with two sectors of “zero” and double zero) at a rate of $5 per color (black): 18\38 x (+5$) + 20\38 x (-5$) = -0,263.

As you may have noticed in both of these examples, the value of the mathematical expectation has a sign “-“, which is typical for most casino bets. Negative mathematical expectation in practice means that the longer the game lasts, the more likely it is for the player to lose.

House Edge [institution share] is a value opposite the mathematical expectation of a player and shows what percentage of bets made in the course of the game for a certain period of time is held in favor of the casino. The most popular casino game in the world is the roulette game, with the European roulette advantage of 1 – 36/37 = 2.7% and American roulette advantage of 1 – 36/38 = 5.26% (with two zeros). This means that if you’ve bet a total of $1,000 in roulette over a period of time, there’s a good chance that around $27 (European roulette) and $54 (American roulette) will eventually go to the gambling house. Casinos outnumber casinos in board games (Baccarat, Blackjack or Craps), so they are more likely to win.

As an example, let’s consider what our casino chances are when playing the American version of roulette, the game wheel of which, let me remind you, has 38 sectors (1-36 digits + 2 sectors of zero). Let’s say we bet on a number. The payment of the winnings in this case is made in a ratio of 1 to 36.

  • The probability of winning in this case is 1\38 or 2.63%.
  • Possible winnings of the player (in percentage terms): 1/38 x 36×100 = 94.74%.
  • Percentage of casinos: 100 – 94.7 = 5.26%
  • Mathematical expectation: [(1\38) x 36 (+1)] + [(37\38) x (-1)] = -0.0263
  • That is, with each dollar you put, the gambling house hopes to earn 2.63 cents. In other words, the mathematical expectation of winning a player when playing American roulette at a casino is -2.6% of each bet you place.


You don’t have to be a great mathematician to play at a casino. You don’t even have to be a great mathematician to play casino roulette. You don’t have to be a great mathematician to play casino roulette, you don’t even have to be a great mathematician to play casino roulette. The main thing to understand is that games with higher math expectations are more advantageous for the player, because in them the casino advantage over you is less and, accordingly, the time of your game and the possible amount of winnings increases. Look for games in which the advantage of the player is realized, only in this case you can count on winning a sufficiently long game.

When choosing roulette, give preference to the European version (with one “zero”), as it will have a casino advantage of 2.7%, unlike the American version (with two “zeros”), in which the advantage of the gambling establishment is already 5.26%.

But when you talk about positive and negative mathematical expectations, you should not forget about the fact that there is a dispersion. And the higher it is, the more you will be “feverish” in the game. You will lose in games with the advantage of the player, and at the same time, you can win where the casino has a significant advantage over the math expectations. Remember that all the mathematics of casino gambling correctly works only when the number of attempts is large and, therefore, to achieve in practice the calculated expected values is quite difficult due to limited budget of the player, the size of bets or time of play.