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In the casino at times we don’t think over that gambling is not only an entertaining thing, but an opportunity to have a rest or to earn money. The science of gambling, winning chances, positive expected payoff are those problems, which have been the fore of scientists discussion (more often physicists, mathematicians and economists) for a long period of time.
We would like to propose you several articles, where we will tell you about the most interesting scientific theories, formulae and discoveries in the field of gambling.
Let’s start with Daniel Bernoulli, the Swiss physicist and mathematician, who lived in XVIII century and was famous first of all for his works in hydrodynamics, aerodynamics, kinetic theory of gases, etc. So it was him who first applied the mathematical analyses of theory of relativity and told about the socalled “St. Petersburg Paradox”. St. Petersburg paradox shows the deviation between the expected payoff with its common estimation of people.
The paradox was described by the following example.
Let us suppose that coming into a game with a coin, a player pays a sum of money, and after that tosses the coin up (the probability of each variant is 50%), until a tail appears. The game is over when the tails fall and the player get the winning. The prize is calculated as follows – if the tail fall with the first tossing, the winner gets 20, while the second  21 and so on (ntossing  2n – 1). So the winning in this game grows twice with each tossing, going through the two’s rate – 1, 2, 4, 8, 16, 32 and so on. The aim of this problem is to find the player’s expected payoff. The paradox lies in the fact that the mathematical meaning of this deposit (infinity) is more than any practically observed winning.
St. Petersburg paradox arises in any game of hazard with the doubling bets for example in roulette. The Martingale system works by the resembling scenario of the bets management. The casino visitor needs only to bet on black. While losing he has to double his bet waiting for the ball to stop on the black pit, and his bets will be repaid. All is well, but there are two “buts”.
At first in order to double the bet every time after losing, you need to have quite a large balance in the casino. For example, if red falls 15 times running, while you are betting on black, your initial bet should increase in 32 768 times (215). And if you start from 1 dollar, not just anyone player has an amount in 33 000 dollars. Secondly and mainly each Internet casino sets up such limits, that the player can make 56 steps maximum (for example, in online roulette the bet diapasons on red/black may be 5300, 251 500 and 503 000). So these precautions may destroy all your strategy.