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Cheating formula: why “winning” casino strategies don’t work

All games available to visitors, casinos, roulette, dice based on the laws of chance. And if in poker or blackjack (in Russia the last game is better known as a point), the player’s skill and experience can affect the result, then the chances of fans of other entertainments are equal. Only the casino is guaranteed to win.

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Julian Nedev / Alamy via Legion Media

Roulette

In roulette, the house’s profit is guaranteed by the zero section, and in the American version also by the double zero. The wheel, or “turntable”, is divided into 37 cells, 36 of them contain numbers from 1 to 36, and the last one contains zero (there are 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 from the loss of numbers is much higher than when guessing the color or parity.

via Legion Media

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Andrew Hasson / Alamy via Legion Media

Without the zero box, the probability of winning for a player betting on, say, black would be 18/36, or 50%. But because of one more cell, it is reduced to 18/37. In other words, the institution has an “extra” share of the chance to win — 1/37, that is, 2.7%. In the American version, due to 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 the black, despite the fact that the winnings seem to be generously paid at the rate of 35 to 1. The player has a 36 out of 37 chance of losing, and only 1 in 37 chance of winning. from each ruble bet 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 chances to leave with extra money than to lose the available ones.

Dice, or craps

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

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commercial images / Alamy via Legion Media

It would seem that everything is fair, because the casino does not directly participate in the game at all. Nevertheless, the gambling house remains profitable here too — the size of the bets is determined so that the participants receive a win less than the «put», that is, calculated according to the laws of probability theory. For example, the chances that the combinations 6+6 or 1+1 will fall out on the dice are 1 to 36, but the bet for them is given at the rate of 30 to 1. If the size of the win was proportional to the probability, then the size of the jackpot would be calculated at the rate of 35 k 1. In the same way, the casino underestimates the winnings for other combinations, taking the difference for itself.

One-armed bandits

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

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ADS / Alamy via Legion Media

Once upon a time, «bandits» were mechanical, and by pulling the handle, the player lowered the spring, which spun the reels with pictures. Today, wheels and gears have been replaced by a computer chip, and cherries, lemons, or card denominations are displayed on the screen. As before, a combination of three identical symbols is considered to be the winning one.

Formally, slot machines work honestly and stop the reels, obeying commands from a random number generator. In fact, each «bandit» is programmed to return to the players a certain percentage of the money invested — usually from 80 to 90%, although the percentage is set to 98% in Las Vegas casinos.

There is no contradiction here: the moment when each reel stops is indeed determined by a random number. But the computer does not use the given value directly.

Instead, the machine calculates according to a certain algorithm: it multiplies, divides and translates from the language of numbers into the language of pictures according to pre-compiled tables. And this is where 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 cheat fortune are hundreds of years old. On the Internet, you can get acquainted for free, and sometimes for a lot of money, with dozens of “100% winning strategies” for playing roulette (for some reason, it seems to players that it is the wheel that is the easiest to “hack”). Fighting probability is useless, but people try hard.

Martingale DOESN’T WORK

One of the oldest roulette strategies requires the player to bet on red or black (or odd or even) and double down on a loss. Sooner or later, the player guesses and breaks the bank.

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

At each step, the chances of guessing are 18 to 37 due to zero, so 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 expense 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 size of the maximum bet in all gambling houses is limited. By betting almost the maximum and losing, a person loses the chance to return the money.

Positive progression strategy DOES NOT WORK

Unlike martingale and similar schemes, players who use the so-called positive progression strategies raise their bets after winning and most often lower after losing.

Schemes with a positive strategy do not allow you to lose quickly, but you will also not be able to get rich with their help, because the casino always has more chances, no matter what bets the player makes. The balance when using such strategies looks something like this:

Favorite number DOESN’T WORK

The player bets on the same number all the time, hoping that a 35 to 1 win will cover his expenses. «Strategists» do not take into account that the 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 not play even once — simply because some other number will fall out twice (by the way, the Biarritz system, which is also very popular among casino visitors, is based on this fact).

If amateurs betting on the same number 36 times in a row did a simple calculation, they would become more tight-fisted. Let’s denote the probability that in 36 spins the numbers will never match as

. Let’s choose any number as a favorite and we will «check» it with the falling numbers. The probability that any next spin will give an unpaired number is 1 — 1/37 (since there is also a zero, the denominator of the fraction will be not 36, but 37). The probability that any of the following revolutions will not give a pair again is 1 — 2/37, then 1 — 3/37, and so on. To find out with what probability all numbers in 36 spins will be different, you need to multiply all these probabilities. In general, the formula looks like this:

where ! is the factorial ( m! is the multiplication of all numbers from 1 to m ), n is the number of turns of the wheel.

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Mira / Alamy via Legion Media

Due to the huge denominator, the number will be so small that there is not enough space on the screen of a conventional calculator to show it. For example, for 36 spins the denominator is 285273917723723876056171083405292782327767461712708093041, and the value itself is 0.00000000000000000000000000000000000000000000000000000003505. That is, there is practically no chance that the numbers will never repeat in 36 spins.

The probability that for any chosen number of spins we will get at least one pair is

If we calculate this parameter for a specific number of spins, then with four spins of the wheel, the chances of at least one «twin» will be 15%, with 7 spins — 45 %, and at 18 — already 99.3%!

Oddities

Offensive coincidences

Lottery lovers also often underestimate the power of probability theory. In September 2009, the numbers 4, 15, 23, 24, 35 and 42 were drawn in the Bulgarian National Lottery. Four days later, these six numbers were drawn again. The organizers of the lottery were suspected of fraud, an investigation was carried out, which 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 be the same as the result of any previous draw. The number of pairs of «sixes» that can be made from all draws is calculated by the formula:

where n is the number of draws , and out of a hundred — already 4950. With so many combinations (possible pairs), the probability that some of them will be the same is significant. For it to exceed 50%, it is enough to conduct 4404 draws — in the case of the Bulgarian lottery, this will take less than 43 years.

Matching lottery draws are not uncommon. For example, in 2010, during two draws of the Israeli National Lottery, on September 21 and October 16, the winning numbers were the same.

Biarritz system DOES NOT WORK

The scheme is based on the fact that in 36 rounds of roulette some numbers are likely to come up two or more times. In the classic version of the scheme, players watch the wheel for a while without placing bets. Having found repeating numbers, they begin to consistently bet on them or, conversely, do not bet chips on these numbers.

The Biarritz system has no mathematical foundations: the probability that the ball will land on a certain number does not depend on whether it hit it on previous spins . But intuitively, people associate future outcomes with those that have already happened (“lightning does not hit the same tree twice”), so the scheme is still popular.

Imperfect wheel WORKS

If the roulette wheel works perfectly, the chances of winning at the casino are always higher. But in real life, the ideal is rare, and in the case of roulette, you can make money on it. This is what the English engineer Joseph Jagger did in 1837. He watched the wheels in Monte Carlo and found that one was not perfectly balanced. 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 figured out what was wrong and swapped the wheels around, but the engineer figured out the catch. Then the owners began to rearrange the cells at night, and other numbers turned out to be winning every day. Jagger cut his playing career short and left Monte Carlo with $325,000 — $5 million in today’s money.

There is evidence that a few more people have been able to find imperfect wheels using statistical analysis. Today, it is impossible to openly look for flaws in wheels — casinos do not favor such “researchers”.

Exact calculation WORKS

This method suggests guessing which cell the ball will end up in, based not on probability theory, but on the laws of physics.

With simple equipment, you can set the speed of the ball and the speed of rotation of the wheel by 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, a computer, and mobile phones won at Casino Ritz in London £1.3m. The gambling house filed a lawsuit against the lucky ones, but the court decided that the defendants did not influence the movement of the ball and the wheel, which means that the win is legal.

Slot machine

Loss streak DOES NOT WORK

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Eric Nathan / Alamy via Legion Media

The idea is similar to that of the Biarritz strategy: the chance of winning is especially high after a long series of failures.