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The mathematics of deception: why casinos are always in the black


Gambling has long been an entertainment opportunity for people to combine luck and skill in the pursuit of profitable winnings. There are hundreds of different forms of gambling and each can be determined by a specific mathematical principle.

The fundamental principle in all casino games is the theory of the probability of winning the lottery.

To calculate the chances of profit or loss, you must have an idea of ​​this theory.

How to calculate the probability of winning

Mathematicians who describe the laws of how to calculate the probability of winning relate events by representing algebraic variables, usually “A” - and fractions of a number from 0 to 1. Thus, the occurrence of (P) events (for example, the suit of the king from the deck of cards) occurs in the form P (A). An event that has no chance of occurrence (for example, five aces from a deck of cards) has a chance of zero, while an event that necessarily occurs (a red or black card from a deck of cards without jokers) has a chance of 1.

The probability of winning the lotteryIs the value obtained by dividing the number of methods that can occur by the total possible number of results in this scenario.

For example, if we want to determine the probability of winning a red suit from a deck of cards in a card game, we would divide 26 (the total number of red cards in the deck) by 52 (the total number of cards in the poker deck, ignoring the jokers), it turns out ½ that is a chance of 0.5.

The logic of probability theory in the lottery has certainly been eternal, although actual mathematical research is a relatively new phenomenon. The vast inherent scenarios that existed during the ancient pastime in gambling is the main factor that prompts research in mathematical terms - people wanted to know in more precise details their chances of winning!

Independent and dependent wins in the machine

There are various formulas for the measure of outcome, and of particular importance in determining the formula for use is figuring out whether the incident is independent or dependent.

In close connection with the theory, the probability of winning the lottery represents concepts, both with independent and dependent events, the expected value and limitations. Understanding these concepts and how they are used in various calculations, combined with probability in the game, is often very helpful in developing an optimal bet strategy.

The methods for developing a strategy for obtaining a certain measure of outcome depend on their dependence. An independent event is based on the results of another and is not affected by the outcome, while the dependent is the opposite of the independent and the result affects or depends on the results.

As an example of independent events - dice. People often think that if they stick with a certain set of numbers for a long enough time, the chances of these numbers improve every time. This is a bad practice, each lottery draw is an independent event, that is, previous lottery draws do not have an effect on subsequent ones. Thus, if you plan to win the lottery, the chances are the same, regardless of whether you choose new numbers every time or stick with the same set of numbers!

The mathematics and probability of winning poker represents related events, as the number of cards decreases each time the cards are dealt and the chances of a certain card falling out of the deck increase each time.

Three basic principles underlie casino games: determining the probability of winning a game, the expected value, and the volatility index. Understanding these concepts will help you understand how theory works and how people outperform competitors.

As a rule, experienced players assess the risk of each round based on the mathematical properties of probability, the chances of profit, the expected value, the volatility index, the duration of the game and the size of the bet. These factors numerically give a picture of the risks and tell the player whether to bet.

It is also useful to know about such things as interest payments and functions, as well as erroneous concepts contained in the player’s misconceptions.

Thus, the probability theory of winning can offer tips on some of the most popular casinos, such as roulette, craps, blackjack, poker, bingo, keno, slots, as well as sports betting.

Gambling is the art and science of game theory: only synthesis will win.

No fraud, or why the casino is always a plus

All casino games - roulette, craps, cards, slot machines - are based on the laws of chance. And if in poker or blackjack (in Russia the game is known even to people far from the casino as a point), the skill and experience of the player can affect the result, then the chances of fans of other entertainment are equal. Only one player has guaranteed winnings - the casino.

In roulette, the profit of the institution is guaranteed by the zero section, and in the American version, also double zero. The wheel, or “pinwheel,” is divided into 37 cells, 36 of which have numbers from 1 to 36, and the last one has zero (38 cells in the USA, of which two are zero). You can bet on specific numbers or groups of numbers or on “equal chances”: black-red and even-odd. The profit margin for falling numbers is much higher than for guessing color or parity.

If there were no zero cells, the probability of winning for a player who put, say, on black, would be 18/36, or 50%. But due to another cell, it is reduced to 18/37. In other words, the institution has an “additional” share of the chance of winning - 1/37, that is, 2.7%. In the American version, because of the second zero, the discrepancy is twice as large and amounts to 5.4%.

When a person bets on a specific number, the gambling house also remains in positive territory, despite the fact that the win seems to be generously paid at the rate of 35 to 1. The player's chances to lose are 36 out of 37, and the chances of winning are only 1 out of 37. That is, from each the ruble put on a specific number, the casino will receive

or the same 2.7%. This does not mean that players are always in the red, but they have much less chance of leaving with the extra money than losing the existing ones.

Dice, or Craps

The rules of the game are simple: a player (shooter) rolls two dice, and if the sum of points on them is 7 or 11, he wins, if 2, 3 or 12, he loses. When a different amount falls on the dice, the shooter throws them to a winning or losing combination. The rest of the participants make bets, trying to guess how the bones will lie.

It would seem that everything is fair, because the casino generally does not directly participate in the game. Nevertheless, the gambling house remains profitable here - the size of the bets is determined so that the participants receive a win less than the “due”, that is, calculated according to the laws of probability theory. For example, the chances that 6 + 6 or 1 + 1 combinations fall on the dice are

but the bet for them is issued at the rate of 30 to 1. If the size of the winnings were proportional to probability, the size of the jackpot would be calculated at the rate of 35 to 1. In the same way, the casino underestimates the winnings for other combinations, taking away the difference.

"One-armed bandits"

Casinos are primarily associated with roulette and poker, but, according to statistics, 61% of visitors to gambling houses spend time fighting with “one-armed bandits” (data from the American Gaming Association for 2013). The rules of the game on the machines are extremely simple, and a frivolous minimum bet makes them accessible even to the poorest players.

Once upon a time, the "bandits" were mechanical, and, pulling the handle, the player released the spring, which unwound the reels with pictures. Today the wheels and gears were replaced by a computer chip, and cherries, lemons or card values ​​are displayed on the screen. As before, a combination of three identical pictures is considered to be winning.

Formally, slot machines operate honestly and stop the reels, obeying commands from the random number generator. In practice each "bandit" is programmed to return to players a certain percentage of the money invested - usually from 80 to 90%, although a share of up to 98% is established in the Las Vegas casino.

There is no contradiction here: the stopping time of each drum is really determined by a random number. But the computer does not use the returned value directly. Instead, the machine calculates according to a certain algorithm: it multiplies, divides and translates from the language of numbers to the language of pictures according to pre-compiled tables. And it is here that the percentage of winning results is laid: by changing the parameters of the table, you can make the “bandit” more or less “generous”.

Ferris Wheel Strategies

Attempts to deceive fortune are not one hundred years old. On the Internet, you can get acquainted with dozens of “100% winning strategies” for playing roulette for free, and sometimes for a lot of money (for some reason, it seems to the players that it’s easiest to “hack” the wheel). Fighting probability theory is futile, but people are trying hard.


One of the oldest strategies for playing roulette requires the player to bet on red or black (or even-odd) and double the bet when losing. Sooner or later, the player guesses and breaks the bank.

The scheme seems logical, but in reality the total gain will not exceed the size of the original bet. Let the player bet on black and guess at the sixth turn (players say back). Then his balance looks like this:

At every step, the chances of guessing are

due to zero, therefore, with a sufficiently large number of spins, the player is in the red. In addition, martingale lovers often have to make many attempts and double the cost each time. If the money runs out before the “strategist” guesses, he will lose a huge amount. Finally, casino owners are well aware of martingale, and the maximum bet amount in all gambling houses is limited. Having set almost the maximum and losing, the person loses the chance to return the money.

Positive progression strategyDOES NOT WORK

Unlike lovers of martingale and similar schemes, players using the so-called strategies with positive progression raise bets after winning and most often lower after losing. Schemes with a positive strategy do not allow you to lose quickly, but you won’t get rich with them either, because the casino always has more chances, no matter what bets the player makes. The balance when using such strategies is approximately as follows:

Favorite numberDOES NOT WORK

The player bets all the time on the same number, hoping that a win of 35 to 1 will cover his expense. "Strategists" do not take into account that numbers fall out evenly only with an infinitely large number of revolutions. And in a real game with a high probability for 36 spins, the selected number will never play - simply because some other number will drop out twice (by the way, the Biarritz system, which is also very popular among casino visitors, is based on this fact). If fans 36 times in a row to bet on the same number made a simple calculation, they would become more tight-fisted.

Let us denote the probability that for 36 spins the numbers will never coincide, as we Choose any number as a favorite and we will “compare” it with the drop-down numbers. The probability that any next spin will give an unpaired number is

(since there is still zero, in the denominator of the fraction there will be not 36, but 37). The likelihood that any of the subsequent turns will again not give a pair is

on and on and on. To find out with what probability all numbers for 36 spins will be different, you need to multiply all these probabilities. In general, the formula looks like this:

Where ! - factorial (m! Is a multiplication of all numbers from 1 to m), n - the number of turns of the wheel.

Due to the huge denominator, you get such a small number that there is not enough space on the screen of a conventional calculator to show it. For example, for 36 spins the denominator of the fraction is 285273917723723876056171083405292782327767461712708093041, and the value itself is 0.000000000000000000000000000000000000000000000000000000003505. That is, there are practically no chances that for 36 spins the numbers will never repeat.

The probability that for any selected number of revolutions we get at least one pair is equal. If you calculate this parameter for a specific number of spins, then with four turns of the wheel, the chances of at least one “twin” will be 15%, at 7 revolutions - 45%, and at 18 - already 99.3%!

Biarritz SystemDOES NOT WORK

The scheme is based on the fact that in 36 rounds of roulette game some numbers are likely to fall out two or more times. In the classic version of the scheme, players watch the wheel for a while without making bets. Having discovered duplicate numbers, they begin to consistently bet on them or, conversely, do not put chips on these numbers.

The Biarritz system has no mathematical foundations: the probability that the ball will stop at a certain number does not depend on whether it hit it on previous spins. But intuitively, people associate future outcomes with what has already happened (“lightning doesn’t fall into the same tree twice”), which is why the scheme is still popular.

Slot machine

Series of failuresDOES NOT WORK

The idea is similar to the Biarritz strategy idea: the chance of winning is especially high after a long series of setbacks. Subconsciously, it seems to a person that you can’t lose all the time and after a black line he will certainly break the bank. The creators of slot machines spur this hope: the "bandits" are programmed with an increased frequency to give out winning combinations one level higher or lower than the main line. The player sees that the drum “almost got into a spin”, and again and again throws the tokens into the coin acceptor.

Wheel imperfectionWORKS

If the roulette wheel works perfectly, the chances of winning at the casino are always higher. But in real life, ideal is rare, and in the case of roulette you can make money on it. Which is what the English engineer Joseph Jagger did in 1837. He watched the wheels in Monte Carlo and found that one balanced imperfectly. Nine numbers - 7, 8, 9, 17, 18, 19, 22, 28, 29 - fell out more often than others.

Jagger started betting on sticky numbers and won $ 370,000 in four days. The owners realized what was happening, and swapped the wheels, but the engineer saw through the catch. Then the hosts began to rearrange the cells at night, and other numbers turned out to be winning every day. Jagger interrupted the player’s career and left Monte Carlo with $ 325,000 - $ 5 million for today's money.

There is evidence that several more people managed to find imperfect wheels with the help of statistical analysis. Today it’s impossible to openly look for wheel defects - the casinos don’t favor such “researchers”.

Accurate calculationWORKS

This method suggests guessing which cell the ball will be in, based not on probability theory, but on the laws of physics. With the help of simple equipment, you can set the speed of the ball and the speed of rotation of the wheel, directly measuring them. By comparing these values, it is easy to calculate when and where the ball will stop. In 2004, three players armed with a laser scanner, computer, and mobile phones won £ 1.3 million at Ritz Casino in London. The gambling house filed a lawsuit against the lucky ones, but the court decided that the defendants did not affect the movement of the ball and wheel, which means that the win is legal.

Offensive coincidences

Lottery lovers also often underestimate the power of probability theory. In September 2009, the numbers 4, 15, 23, 24, 35, and 42 fell out in the national lottery of Bulgaria. Four days later, these six numbers fell out again. Lottery organizers were suspected of fraud, an investigation was conducted that established that everything was fair. The calculation shows that the probability of repeating the six numbers in the Bulgarian lottery, which has been held for 52 years twice a week, is very high.
The result of each draw may coincide with the result of any of the previous ones. The number of pairs of “sixes” that can be made up of all the draws is calculated by the formula:

Where n Is the number of draws.

Of the two draws, you can make only one pair, from three - 3, from four - 6, from five - 10, and from one hundred - already 4950. With so many combinations (possible pairs), it is likely that some of them will be the same, significant. For it to exceed 50%, it is enough to conduct 4,404 draws - in the case of the Bulgarian lottery, it will take less than 43 years. Matching lottery draws is not uncommon. For example, in 2010, during the two draws of the Israeli national lottery, September 21 and October 16, the same numbers were winning.