How Is Math Used in Gambling?

How Is Math Used in Gambling?

There is a great variety of gambling entertainment. All games of this type have one keynote characteristic – winnings depend on a chance rather than the skills of the player. However, players still can define the probability of a certain combination, as well as find out their chances to win. All of it is possible thanks to math calculations. Read more about the use of math in gambling in the article below.

Math and gambling: a little of the history

Gambling dates back to a long time ago. In ancient times, craps (or a game of dice) was widespread in India and Greece. At that time, people used astragals (animal bones) instead of cubes.

In the Middle Ages, people wondered how many possible outcomes craps had and how many ways allowed to get those combinations. In 960, the French bishop Wiboldus wrote a work, where he attempted to answer one of these questions. He calculated that the toss of three dice gave 56 possible game outcomes. However, it turned out later that this number did not reflect the real number of equally probable possibilities. The reason is that each of 56 possible game outcomes can be attained by summing up different numeral combinations. For example, the bishop stated that figure 4 came with a combination of 2 + 1 + 1. In fact, there are three combinations that give figure 5 when summarized 2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2.

In 1494, mathematician Fra Luca Bartolomeo de Pacioli published a book, where he described how to divide a joint bet between two participants if the game ended ahead of time. The author offered to divide the bet proportionally to points of competitors. Later it appeared that he solved that task incorrectly.

In the 15th century, mathematician and engineer Gerolamo Cardano wrote The Book on Games of Chance, which presented research on the mathematical theory of gambling. He was the first to get close to the general concept of probability theory. He stated that there was one common calculation rule: to take into account the total number of possible outcomes and the number of ways that can bring those results. Then it was necessary to find the ratio of the latter number to the number of the rest possible outcomes.

Besides, Blaise Pascal and Pierre de Fermat made a significant contribution to the development of the probability theory. In their correspondence, they managed to solve the task of dividing the bet between two participants correctly for the first time ever (which Pacioli failed to solve earlier). They offered solutions that contained elements of using mathematical expectation, as well as theorems on addition and multiplication of probabilities. In the end, a number of their theses laid the foundation of the probability theory.

Later, such well-known mathematicians as Christiaan Huygens, Jacob Bernoulli, Abraham de Moivre, and others raised the theme of using math in games of chance.

Probability theory and
gambling: how does it work?

The theory of probability is a branch of mathematics that studies regularities of random phenomena. Probability is the extent to which an event is likely to occur.

With mathematical approaches, one can calculate the probability of getting a certain card, the chances of the player to win in a game of chance. Calculations are possible for such games as roulette, craps, blackjack, poker, lottery, etc.

Let’s see in detail how to use math in games of chance.

Dependent and independent events:
what influences the game outcome?

Events are called independent if the occurrence of event A does not influence the probability of event B occurring.

For example, if you toss up a coin twice, the outcome of the second toss will not depend on the first one. This suggests that occurred events do not influence each other in any way. In this case, to calculate the probability of the coin landing on a certain side, you can use the formula: (1/2) × 2 = ¼ or 25%.

The event is called dependent if apart from random factors its probability also depends on the occurrence or nonoccurrence of another event.

Here is the example of how to calculate the probability of each of three randomly taken cards from the pack to be ace. The standard so-called French pack contains 52 cards including four aces. The chance of getting an ace from the first attempt is 4 to 52. If the first taken card is an ace, 51 cards will remain in the pack with three aces left. Then the probability will be 3 to 51. If the second taken card is also an ace, the probability of getting the third card of the same value will be 2 to 50.

It is important to understand that in the case of dependent events, each new action influences the outcome of the next action. In the abovementioned case, each subsequent extraction of a new card influences the probability of the outcome of the next event.

The probability of the positive outcome of the event, when each of three randomly taken cards turns out to be an ace, is calculated according to the following formula: 4/52 × 3/51 × 2/50 = 0.000181.

Mathematical expectation

The mathematical expectation is one of the most important notions in the probability theory. It is defined as the average probabilistic value of a random variable. In gambling, this notion denotes the sum that the player can win or lose upon the condition that he or she is placing similar bets for a long time.

Mathematical expectation can be positive or negative. For example in roulette, the ball lands on black more often than on red in percentage terms. Consequently, when you bet on black, the mathematical expectation is positive. When you place a bet on red, it is negative. Besides, this parameter can equal to zero. This happens when you toss up a coin. In this game, heads and tails land upward with the same probability.

The mathematical expectation is also used in betting. In this field, it is defined as a sum that the punter may win or lose if he or she bets with the same odds for numerous times.

To calculate mathematical expectation, you can use the following formula: to multiply the probability of the positive outcome by the sum of possible winnings. The probability of a negative outcome is multiplied by the sum of possible losses. Then you have to deduct the amount obtained in the second operation from the first value.

Here is the example of calculating mathematical expectations in sports betting.

Let’s say that in the match between Dynamo and Shakhtar, the probability of Kyiv team winning is 1/3.30 (or 0.303), the chances for Donetsk team winning is 1/2.18 (0.459), and the probability of tie score is 1/3.95 (0.253). If the probability of Dynamo’s victory is 0.303, the team’s chances to lose are 0.459 + 0.253 = 0.712. Let’s say that you decide to bet on Dynamo 1000 UAH. With the given odds, the possible win amount is 2300 UAH.

Now we use the obtained values in the abovementioned formula: 0.303 × 2300 – 0.712 × 1000 = -15.1. As a result, we managed to define that the average loss amount is 15.1 UAH for this kind of bet.

Note that in the case of the continuous play, the gambler may win only in the case of positive mathematical expectation. Besides, an important thing is that hardly any games of chance have positive mathematical expectation. The reason is that a certain percentage of bets goes to the casino’s budget. Consequently, regardless of the game outcome, the player loses some money.

How to calculate
chances to win?

With the help of math, you can calculate not only the probability of getting a certain card or a loss. You can also calculate chances to win a game of chance.

For example, using mathematical operations, you can define the probability of winning a lottery. This parameter depends on two variables: the total number of figures available in the game and the number of figures you need to guess. To calculate chances to win, use the following formula:

x figures from n = (n) / (x) = n × (n – 1) × (n – 2) × (n – 3) … × [n – (x -1)] / 1 × 2 × 3 × 4 × … x,

  • where n is a total number of figures;
  • x is a number of figures you need to guess.

Let’s see the example of a lottery where you need to guess 6 figures out of 45 to win. The total number of possible combinations is calculated as follows:

45 × 44 × 43 × 42 × 41 × 40 / 1× 2 × 3 × 4 × 5 × 6 = 8 145 060

The obtained number indicates that the probability to win the lottery is 1 to 8 145 060.


Using mathematical operations, players can increase their chances to win. It is important to note that many casinos discourage such an approach to games. Some gambling houses ban players that were caught calculating the cards. For this reason, you should be careful when you use math for gambling.

You can read about myths related to slot machines here ►►►

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