Mathematics and casino

Mathematics and CasinoIn any undertaking, two-thirds depends on reason, one-third on chance. Increase the first fraction and you are faint-hearted. Increase the second and you are foolhardy.

In this article we will cover the basic principles on which is based the activity of gambling houses, as well as how they get profit and how lucky they may be. Let us start with the basic mathematical laws on which is built the gambling. What is the link between mathematics and casino? Many casino games were created and developed by mathematicians. May we use their weapons to get advantage over a gambling house?

A bit of history

In 1526 the Italian mathematician Geralomo Cardano for the first time tried to describe the dice game with the help of mathematics in his "Book on Games of Chance". By having a study on his own gaming practice, he tried to develop and theoretically justify the system of recommendations of the stakes management. In fact, he was the one who worded the definition of probability:

"Probability theory is concerned with determining the relationship between the number of times some specific given event occurs and the number of times any event occurs."

Later, at the end of 16th — beginning of 17th centuries, the mathematics analysis of dice game was continued by Galileo Galilei and Blaise Pascal. They started doing this at the request of their friends who were big gamblers with huge spendings on gambling. It must be acknowledged that the science of probability, according to the history, grew out of mercantile problems of the gamblers.

It is commonly believed that at that time a whole area of mathematics was born, wholly dedicated to probabilities. The next step in this direction was made by the Dutch mathematician Christiaan Huygens, who published a book in the middle of the 17th century "On Reasoning in Games of Chance" ("De Ratiociniis in Ludo Aleae"). The further development of the probability theory was done in the writings of the great mathematicians of the 18th-19th centuries – Jacob Bernoulli, Poisson, Laplace, Moivre and others. Very soon the new theory is widely used in the areas which are quite different from gambling.

Mathematics at casino games

CoinGambling and probability theory, how do they both work? Lt Let us see if there is any link between gambling and mathematics. At a coin toss any of its sides has the same probability of dropping. So we have one of two outcomes – heads or tails. The probability of getting heads is ½ (50%), so half of the tosses will be heads.

The probability is how often an expected outcome may happen, and it is represented as a ratio of expected outcomes of the total possible outcomes at a large number of retries within a long period of time.

The probability of an outcome reflects the quantitative possibility of that outcome. If it is equal to zero, that outcome may not happen at all. If it is equal to 1 (100%) – the outcome will definitely happen. You may find practical advice on the use of mathematical calculations at a casino on the following page:


A standard deck has 52 cards, including 4 ACEs. The probability of getting an ACE is: (4 / 52)  * 100 = 7,69%. The European roulette has 37 cells on the wheel: 1-36 – numbers (18 reds and 18 blacks) and the zero colored green.

  • The probability of getting any of the numbers (1/37)*100=2,7%.
  • The probability of getting a red number – (18/37)*100=48,6%.
  • The probability of getting the dozen – (12/37)*100=32%.

Win/loss ratio

Winning and loosingIf talking about the mathematical probability of a win at a casino, it is often viewed as a loss/win ratio, so we take the win/loss ratio.

  • When rolling two dice, there are 36 outcomes (a cube has six faces, each of which can match any of the faces of the other cube).
  • Consider the probability of getting a seven from a two dice rolling. It may happen in the following outcomes: 3 and 4; 5 and 2; 6 and 1; 4 and 3; 2 and 5; 1 and 6. Therefore, 5 (out of 6) outcomes are negative and just one positive. The loss/win ratio in this case is 5 to 1.
  • The given example consists of mutually exclusive outcomes: you get the numbers which make a 7, or you do not get the numbers which make a 7. These are called mutually exclusive outcomes/events if, under any circumstances, they may not happen at the same time.

Opposite events:

  • The opposite of the event is its complement. The complement of heads is tails, the complement of the red color is black, the complement of odd  is even. The probability of all potential outcomes is always equal to 1.
  • For example, at getting a random card from the deck, there will be either hearts [13 / 52, or 25%], or any other suit [39 / 52, or 75%]. So we have: 13 / 52 [25%] + 39 / 52 [75%] = 52:52 = 1 [100%].
  • Let us see what is the probability of getting hearts or spades. These events are mutually exclusive and the probability of each of them is 13 to 52. The chance of getting hearts or spades is 13/52 + 13/52 = 26/52 = 1/2  [50%]
  • The casinos are based on the same mathematical laws and principles.

Independent events

If the probability of an event outcome does not affect the probability of the other, these events are called independent. Let us toss the coin twice. The second outcome absolutely does not depend on the first one. Both of these events have no affect on each other, so they are independent.

  • The probability of getting tails in one of the coin toss is: (1/2)2 = 1/4 (or 25%)
  • The probability of getting tails ten times in a row is: (1/2)10 = 1/1024 (or 0.098%)
  • In one of the casinos in Las Vegas was presented  a pair of dice. The inscription reads that the uniqueness of these dice is thanks to their 28 straight passes which once happened. Note that the probability to get 28 straight passes in DICE game is (0,493)28, or approx. 1 out of 400 million. So the casino recognizes the mathematical uniqueness of this event.

Dependent events

Mathematical probability of drawing four acesLet us assume we get three ACEs from a cards deck. The chance of getting an ACE first is 4 to 52. If the first card is an ACE, then we have 3 ACEs remaining and the number of cards in the deck now is 51. In this case the probability of getting another ACE is 3 to 51. The same for the third ACE – 2 to 50 (50 cards, 2 ACEs in the deck).

  • Let us have a mathematical calculation of the positive outcome of the given event: 4/52 * 3/51 * 2/50 = 0,000181, so 1 positive outcome out of 5525 tries.
  • Each of the three events consistently affects the probability of the outcome of the next one, so the events are dependent.
  • If each card we get is returned to the deck, the events are independent, consequently, the probability of getting 3 ACEs is: 4/52 * 4/52 * 4/52 = 0,000455, so 1 positive outcome out of 2197 tries.
  • Each of the three events consistently affects the probability of the outcome of the next one, so the events are dependent.

Mathematical expectation (Expected Value)

The essence of understanding the mathematical expectation (also known as: player expectation, expected value) is quite simple. Speaking in a clear way – it is the amount of money you can win or lose within a long period of time, provided that you are going to make the same bet.

You may want to calculate the value of mathematical expectation using the following formula:

МО = (the number of positive outcomes [wins] / the number of possible outcomes) * the winnings amount + (the number of negative outcomes [losses] / the number of possible outcomes) * the bet amount. Many of you will see this as a Chinese inscription, but it is actually quite simple.


You wager 1$ on hearts to be the first card. According to the probability theory, the positive outcome (you get the hearts and you win +1$) will happen with a probability of ¼, the negative outcome (you get another suit and you lose 1$) will happen with a probability of ¾.

Let us calculate the mathematical expectation using the above formula:
МО = 1/4 * (1$) + 3/4 * (-1$) = - ½$

Thus, within a long period of time your loss will be 50 cents for every dollar wagered, so, according to the mathematics, the 4 attempts will make you lose three times, 1$ each (you get a loss of 3$) and win a single time - 1$.

Mathematical expectation at roulette

RouletteLet us calculate the mathematical expectation at roulette (the American one with two zero sectors: zero and double zero) when wagering 1$ on color (black): 18/38 * (+1$) + 20/38 * (-1$) = -2/38 = -0.0526 (or -5.26%).

As you have probably notices, in both of the given examples, the value of mathematical expectation has a "-" (minus), it is typical to most casino wagers. The negative mathematical expectation means: the longer the game, the greater the probability of loss.

Casino advantage (House Edge) [house percentage] is the value which is opposite to the mathematical expectation of the player; it is the casino advantage (percentage) over the player. The casino advantage in European roulette is 1 - 36/37 = 2,7%, in American roulette is 1 - 36/38 = 5,26% (thanks to two zero sectors). This means that if you wager $1000, the probability of losing 27$ (in European roulette) and 54$ (in American roulette) is quite high. At board games the casino advantage is lower (Baccarat, Blackjack or Craps).

Let us take American roulette again, which has 36 numbers and 2 zero sectors. Suppose we wagereg on a number. In this case the probability of win is 1 to 36:

  • The probability of win: 1/38 or 2,63%;
  • The possible winning (the percentages to wager): 1/38 * 36*100 = 94.74%;
  • Casino percentage: 100 – 94,7 = 5.26 %;
  • The mathematical expectation: (1/38) * 36 (+1) + (37/38) * (-1) = -0,0263.
  • So, from each wagered dollar the gambling house may get 2,63 cents. In other words, the mathematical expectation in American roulette is -2.6% from each of your bets.

The mathematical dispersion at casino games

In mathematics, the dispersion is a statistical measure which tells you how measured data vary from the average value of the set of data. In our case, it is the risk degree. Being used in gambling, the dispersion is the deviation degree of the outcome from its mathematical expectation. The dispersions makes the game unpredictable, you either win or lose.

The gambling houses exist thanks to the dispersion: any outcome would be calculated mathematically. The disperions is neither a positive nor negative factor, it exists by itself as an objective reality. To some extent it compensates the negative mathematical expectation, allowing the player to win (at a short distance). At the same time it does not allow to create a working system which guarantees winnings at a long distance.

It should be noted that when wagering on "color" the dispersion in roulette is almost absent. In practice, however, there are records of 15 straight droppings of the same color. Learn more about dispersion in the following articles:

The law of large numbers

The law of large numbersIf the probability of events are identical, it does not mean that we will get from now on such an outcome. Suppose we toss ten coins at the same time. It is logical to expect 50% of tails. However, the probability of getting 60% or above is quite high. This is thanks to dispersion we talked about earlier.

By tossing a coin ten thousand times, we get a balanced expected value (50%). The probability of getting 60% or a larger number of tails at a random toss of 10 coins = 0,377. Let us get the same for one hundred coins. The probability of getting 60% of tails equals to 0,028, or approx. 1 out of 35. If tossing 1000 coins, to get 60% or a larger number of tails is quite impossible. The probability of this event equals to 0.000000000136 (less than 1 out of 7 billion). We will not get the 50% of tails, but the more coins we have, the closer we are to the average value (50%).

That is how the "law of large numbers" works: the accuracy of the expected outcome ratio (according to the probability theory) is higher when we have a larger number of events. By using this law, you may accurately only predict the outcome of a series of similar events. Although the outcome of each individual event is unpredictable, it is balanced at a long distance.

How to get a positive mathematical expectation at casino?

On our site we have a list of strategies and recommendations which should be used to get a positive mathematical expectation at Netent slots. It is based solely on mathematical calculations, taking into consideration the payout percentage of each of the Net Entertainment games and the wagering requirements of the bonuses. Find out more on the following page:


There is no need to be a great mathematician to play at casino. You may not even calculate the mathematical expectation and dispersion – it was done long before, you may use the existing information. The main thing is to realize that the games with a high value of mathematical expectation (especially the positive one) is more profitable for the player, by getting it you have an advantage over the casino. Play at European roulette (with a single zero sector), here the casino advantage is 2,7%, while at American roulette (with two zero sectors) it is 5,26%.

Recommend you to keep an eye on online casino, where you may find roulette with no zero sector (Zero edge Roulette). It is the most profitable roulette. In this case,  the casino advantage is lowered from 2,7% (European roulette) to 0. The truth is it is compensated by a number of rules which I strongly recommend to read carefully before starting the game. Its percentage the casino takes either as a fee applied on your bet or on your winnings. I guess the last one is the best choice.

In any case, we should not forget about dispersion. The higher it is, the more stressful the game. Remember, the mathematics in gambling works correctly only at a large number of tries; so, calculating the expected values is quite difficult, due to the limited budget, the bet size or play time.

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