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Mathematics and Canadian online casino

Mathematics and casinoIn any business, two-thirds depend on a reason, one-third on a chance. Increase the first fraction, and you are faint-hearted. Increase the second, and you are foolhardy.

In this article, we will talk about the connections between mathematics and casinos and how they are always winning and do they lie on luck. You need to know that most of the casino games were developed by Mathematics, so do not be surprised if you realize that not everything is luck in gambling. Can we use their calculations to get an advantage over the casino? Read the article further and answer yourself.

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A piece of History

In 1526 Italian mathematician Geralomo Cardano was the first one who wrote about mathematics at the dice game in his book "Book on Games of Chance." After a lot of time spent in practice, he tried to make bet management recommendations based on his experience:

"Probability theory is concerned with determining the relationship between sometimes when some specific given event occurs, and several times an event occurs."

Later, at the end of the 16th — beginning of the 17th century, Galileo Galilei and Blaise Pascal continued the same mathematical analyses for the dice game. A friend of then (big gambler) asks them to help him. You need to know that the theory of probability grew because of the gambling problems of a player. 

It is commonly believed that at that time, a whole new branch of mathematics was born, wholly dedicated to probabilities. The next step in this direction was made by 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"). After that, the further development of the probability theory was done in the writings of many great mathematicians of the 18th-19th centuries – Jacob Bernoulli, Poisson, Laplace, Moivre and others. Very soon, a new theory became widely used in the spheres which are entirely different from gambling.

Mathematic at the slots

CoinHow do the probability and gambling theories work? First, let’s try to find out if there is a connection between mathematics and gambling. At a coin toss, any of its sides have 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. It is represented as a ratio of expected outcomes of the total possible outcomes at many retries within an extended period.

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 happen. Find more mathematical recommendations in the following link:


A standard deck has 52 cards, including 4 ACEs. The probability of getting an ACE is: (4 / 52) * 100 = 7,69%. European roulette has 37 cells on the wheel: 1-36 – numbers (18 reds and 18 blacks), and zero is in coloured 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 dozen – (12/37) *100=32%.

Win/loss ratio

Winning and loosingIf we are talking about win/loss ratio and mathematical expectations, the first thought is connected with the casino, so let’s try to explain it:

  • While 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).
  • The probabilities of getting seven from the two dice are 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 that make 7, or you do not get the numbers that make 7. These are called mutually exclusive outcomes/events if, under any circumstances, they may not happen at the same time.

The opposite side

  • Opposite the event - it's a compliment. The complement of heads is tails, the complement of red is black and odd is even. So the probability of all potential outcomes always equals 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 the probability of getting hearts or spades is. 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%]
  • Remember that all casinos are based on the same principles and laws.

Independent event

If the probability of an event outcome does not affect the probability of another, these events are called independent. For example, let us toss the coin twice. The second outcome does not depend on the first one. Therefore, both of these events do not affect each other, so they are independent.

  • The probability of getting tails in one of the coin tosses 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, a pair of dice was presented. The inscription reads that the uniqueness of these dice is thanks to their 28 straight passes, which once happened. Note that the probability of getting 28 straight passes in a DICE game is (0,493)28, or approx—1 out of 400 million. So, the casino recognizes the mathematical uniqueness of this event.

Depended on event

Mathematical probability of drawing four acesLet us assume we get three ACEs from the card deck. The chance of getting an ACE first is 4 to 52. If the first card is 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)

One of the simplest things for the players in gambling is the mathematical expectation (expected value). Of course, all the described examples are very confusing for the normal brain, but for the gambler's mind, it is all about money; that's why they understand the examples much more comfortable because they connect the numbers with money.

The next formula about the value of mathematical expectation can be useful for you.

МО = (the number of positive outcomes [wins] / the number of possible outcomes) * the amount of the winning + (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 quite simple.


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

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

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

Roulette’s mathematical expectation

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

As you have probably noticed, in both given examples, the value of mathematical expectation has a "-" (minus), which is typical to most casino wagers. Negative mathematical expectation means the longer the game, the higher the probability of loss.

Casino advantage (House Edge) [house percentage] (RTP) is the value that is opposite to the mathematical expectation of the gambler; it is the casino advantage (percentage) over the player. The casino advantage in European roulette is 1 - 36/37 = 2,7%, in American - 1 - 36/38 = 5,26% (thanks to two zero sectors). This means that if you bet 1000 CAD, the probability of losing 27 CAD (in European roulette) and 54 CAD (in American roulette) is quite high. At table 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 bet on a number. In this case, the probability of a win is 1 to 36:

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

Dispersion and mathematic at slots

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

The gambling houses exist thanks to the dispersion: any result would be calculated mathematically. The dispersion is neither a positive or negative factor, and it exists by itself as an objective reality. To some extent, it compensates for negative mathematical expectations, allowing the player to win (at a short distance). At the same time, it is impossible to create a successful strategy for winning in the long term.

It should be noted that when betting on "colour," the dispersion in roulette is almost absent. In practice, however, there are records of 15 straight droppings of the same colour. 

Law of large numbers

The law of large numbersIf the probability of events is identical, it does not mean that we will get such an outcome from now on. For example, suppose we toss ten coins at the same time. Then, it is logical to expect 50% of tails. However, the probability of getting 60% or above is quite high. This is thanks to the 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 equal 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 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 predict only the outcome of a series of similar events. Although the outcome of each event is unpredictable, it is balanced at a long distance.

How to get a positive mathematical expectation at the casino?

On our site, we have a list of strategies and recommendations that can be used to get positive mathematical expectations. It is based solely on mathematical calculations, considering the payout percentage of each of the slots and the betting requirements of the bonuses.


Playing at a Canadian online casino is very easy; you don’t have to be very good at mathematics. You don’t even need to try to calculate the mathematical expectation or dispersion; it was done a long time ago. However, you need to realize that the casino games that can bring you a positive mathematical expectation need to be your first choice when starting your gaming session. For example, play at European roulette (with a single zero sector), here the casino advantage is 2,7% when at American roulette (with two zero sectors), it is 5,26%. Do not forget to find a game according to your playing style. If you are looking for a game without risking too much, find games with lower dispersion; higher dispersion games can increase stress. Remember that mathematic in casinos can be calculated for the long term and can happen everything in a short period.

Where to play?

  • Playing at Fastpay casino, the mathematical expectation of receiving your requested withdrawals (if you are not breaking the rules) is 100%. The most recommended gambling site.

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Written by Kamen Valev
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