Many casino games were invented and developed by mathematicians. Can they use their weapons to gain an advantage in the gambling house? Many think mathematics is useless, but you can get rich with specific knowledge. It’s about the casino. This article will analyze how mathematics is related to casinos.

Mathematics of casino games

How does gambling work in terms of probability theory? Probability shows how often the result we expect can be achieved and can be represented as the ratio of expected outcomes to the total number of all possible products. It’s a sufficiently long period and with a large number of repetitions.

The probability of an event reflects a quantitative assessment of the possibility of the occurrence of this event. It cannot happen in principle if it is equal to zero. The event will occur when it is similar to one (100%). There are 52 cards in a standard playing deck, including four aces. The probability of drawing one of the aces from the deck is (4 / 52) * 100 = 7.69%. There are 37 cells on the European roulette wheel: 1-36 are numbers (18 red and 18 black) and a green zero mark.

  • The probability of getting any number is (1/37)*100=2.7%.
  • The probability of getting a red number is (18/37)*100=48.6%.
  • Probability of dropping a dozen – (12/37)*100=32%

Opposite events

The opposite of an event is an addition. An eagle’s complement is tails, red is black, and the complement of an even number is odd. The total probability of all potential outcomes is always equal to 1. For example, when drawing an arbitrary card from a deck, either a card of a heart suit [13 / 52, or 25%] or a card of another suit [39 / 52, or 75%] will be selected. Similarly, the probability of picking hearts or no hearts is 13 / 52 [25%] + 39 / 52 [75%] = 52:52 = 1 [100%].

And what is the probability that an arbitrarily chosen card will be a heart or a spade? These events are mutually exclusive, and the likelihood of each of them is 13 to 52. The chance of choosing a seat or spade card is 13/52 + 13/52 = 26/52 = 1/2 [50%]. The same mathematical laws and principles are subject to casino games. In modern online casinos, you can get Dazard bonus codes that help you win. And imagine if you add mathematical calculations to the bonuses. Thus, the probability of winning will be very high, and you will not have to worry about losing money.


Win/Loss Ratio

Speaking about the mathematical probability of winning in a casino, it is often considered as a ratio against a win. That is, the ratio of the number of unfavorable outcomes of an event to the number of favorable ones is taken for analysis. When throwing two dice, there can be 36 possible options (one die has six faces, each of which may coincide with any face of another cube).

Consider the probability of getting several seven when throwing two dice. It can fall out in 6 cases, provided the following digital combinations match: 3 and 4; 5 and 2; 6 and 1; 4 and 3; 2 and 5; 1 and 6. Therefore, in 5 cases (out of 6 throws), the result will be negative and only in one case positive. The ratio against winning in this example would be 5 to 1.

Independent events

If the probability of the outcome of one event does not affect the likelihood of the effect of another, these events are called independent. Let’s flip a coin twice. The result of the second roll is independent of the result of the first roll. Both of these events do not influence each other. 

Mathematical variance in casino games

In mathematics, variance is a quantity’s deviation from its mean value. In our case, this is the degree of risk. In gambling, variance is the degree of variation of the game’s results from their mathematical expectation. Variance brings an element of unpredictability to gambling, providing the possibility of random wins and losses.

Gambling establishments owe their existence to dispersion, without which there would be no gambling in principle: any outcome would be calculated mathematically. Distribution cannot be attributed to either a positive or a negative factor; it exists as an objective reality. To some extent, it compensates for the negative mathematical expectation of the player, allowing him to win. 


Law of Large Numbers

If the probabilities of any events are identical, this does not mean we will receive such a result in any situation. Let’s say we toss ten coins at once. It is reasonable to expect tails to come up about 50% of the time. However, it is realistic to get a figure of 60% or higher. It is a consequence of dispersion. But if you flip a coin ten thousand times, the values will change in the direction of the expected value (50%). The probability of getting 60 percent or more tails on a random toss of 10 coins = 0.377.

It is how the “law of large numbers” works. It says: the accuracy of the ratios of the expected (according to probability theory) results is higher the greater the number of events observed. With the help of this law, only the result of a massive series of similar events can be accurately predicted. And although the outcome of each event is unpredictable, it is maximally averaged over a large sample.


You can be a great mathematician by playing in the casino. You don’t have to count the mathematical expectation and variance – this was done before you, and you can use the ready-made results. The main thing to understand is that games with a high value of the mathematical expectation are more profitable for the player, in which the advantage of the casino over you is less. Remember that all the mathematics of gambling only work correctly in the case of a large number of attempts. It is pretty challenging to achieve the calculated expected values in practice due to the player’s limited budget, the stakes’ size, or the game’s time.