The casino phenomenon attracts a large number of gamblers looking for easy money. Casinos, as true hedonistic institutions, create the false perception that card games, slot machines, or roulette can instantly enrich an individual with a minimal initial investment. This perception is reinforced by success stories of individuals who have already hit the jackpot and won significant amounts of money. However, in reality, these beliefs have no mathematical basis and, on the contrary, contradict the formulas and numbers. In other words, the casino always wins, and the player always loses.
One of the most important cognitive errors associated with the casino phenomenon is Gabler’s fallacy or false Monte Carlo conclusion. The statistical meaning of this error is that the player tends to associate independent events with each other. For example, a player may believe that if the roulette wheel is black three times in a row, then the probability of the following red increases. This is not true because such events are unrelated, and thus the probability of their occurrence does not depend on previous experience. History knows an example in a Monte Carlo casino where the ball stopped on a black field 26 times in a row, causing individuals to be biased about whether the next one would be red and to lose repeatedly.
As a result of the player’s cognitive biases, they unconsciously lose the benefit, contributing to the casino’s winnings. Meanwhile, it should be remembered that few real players in the casino play only one round: on the contrary, the atmosphere of excitement makes them constantly repeat the game. Even if a player was lucky and managed to win a round, it is unlikely that they will finish the game because cognitive errors will make their believe that they will continue to be successful. Consequently, the participant will create a distribution consisting of rounds within a particular type of game. In this sense, Bernoulli’s theorem of large numbers is applicable, allowing us to determine the probability of a favorable outcome occurring a specific number of times. For example, if a casino offers a short game 2 whose goal is to guess which side a coin will fall, heads or tails, then as we know, the probability of either event is 0.5. Suppose it costs $1 to participate, which to an individual might seem like a very favorable price with a prize of $2.
These factors encourage the individual to keep participating (n = 100), which creates a binomial distribution. The probability of winning 50 times out of 100 should be the same since the probability of one outcome is 0.5. However, as the calculation for Bernoulli’s theorem shows, the actual probability of such an event is 0.0796 [1]. In other words, the binomial distribution does not allow the player to win in the long run and creates opportunities to lose.
The player’s long-term loss also contributes to the concept of mathematical expectation, which reflects the weighted average expected value of the sum of all outcomes. This concept is best explained in practice: for example, an untrained player may think that roulette seems to be a fair game because the probability of winning is determined only statistically. Falling red or black sectors are equal probability events. Suppose a player decides to bet $100 on each hand only on red cells. The prize ($200) is won only if the participant can guess — otherwise, the $100 goes to the casino. The basic probability that the red sector will fall, and the player will win is about 47%, given the two green cells on the field. In that case, the mathematical expectation would be -5.26, which makes the game unfair: for every hundred dollars bet, the player loses $5.26 on average [2].
Thus, the game leads to a loss in the long run anyway.
To summarize, the casino’s consistent winnings are ultimately not due to cheating or fake gambling but to mathematical concepts. Gambling is based on statistical rules that make games unfair in the long run. This means that even if a player gets lucky once, they will lose their money in the future because that is what mathematics postulates. For this reason, the casino always wins solely on the laws of mathematics.