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Can the knowledge of mathematics help a casino player to win?

One can often hear that the best piece of advice given by a mathematician to a lover of casino games is an assertion which lies in the fact that the best casino games strategy in gambling games and casino games is complete abstention from participation in them. A lot of mathematicians consider that the most which the theory of probability and the theory of games can give a casino player are the strategies following which he won't lose too much.

It is difficult to predict whether the American mathematician Edward Thorp shared this view, when once spending winter holidays in Las-Vegas, he, having entered a casino, decided to try his luck in the casino game of twenty-one. As it turned out, "Dame Fortune" was extremely unkind to him. We do not know for sure what amount of money this teacher of mathematics of one of American universities lost in the casino that winter night at the end of the 50-s - the beginning of the 60-s of the last century, however, judging by the following events the amount was not small. Otherwise, how can we account for the fact that development of an optimal strategy of this game became for a number of years an "idte fixe" of our hero. Besides, the matter was not only in the quantity of money lost in the casino by the mathematician. Perhaps, Thorp was simply an extremely venturesome person, and his pride both of a gambler, casino player and an expert-mathematician was hurt. Besides, he could suspect a casino croupier of dishonesty, since, as he had noticed, casino cards were not shuffled after each casino game. Though, during the game itself it did not make him very uneasy. However, afterwards, having visited casinos a number of times, he noticed that as the rules did not presuppose obligatory shuffling of casino cards after each game, so it was difficult to accuse a casino croupier of anything. Anyway, he managed to develop a winning strategy in the casino game of twenty-one.

This strategy among other things was based on the same very aspect which had put a defeated mathematician on his guard - casino cards were not shuffled too often. At that, this, apparently, as a rule, was done not because of some evil design, but in order to avoid, so to say, unnecessary slowdowns in the casino game. The results of his studies Edward Thorp put forth in a book published in 1962 (Thorp E.O Beat the dealer. A winning strategy for the casino game of twenty one. - New York: Blaisdell,1962.) which made owners of gambling houses and casinos in the state of Nevada essentially change the rules of the casino game of twenty-one. But let's not ride before the hounds.

In accordance with the casino game rules of twenty-one of that time one casino croupier dealt casino players two cards each out of a thoroughly shuffled pack consisting of 52 cards. Casino players themselves did not show their cards to a dealing casino croupier. At the same time out of two cards taken for himself an official of a casino showed one of them (usually the first one) to casino players. Casino players evaluate their cards according to the following scale. Jacks, queens and kings have a value equal to 10 points, an ace could be assigned either 1 point or 11 points, the value of the rest of the cards coincided with their numerical value (eights had 8 points, nines took 9, and etc). That casino player was considered a winner who had cards on hand with the sum of points closest to 21 from the bottom. At that, having assessed the received cards every casino player (including a casino croupier) had a right to take from a pack or putting it simpler, take a "widow", any amount of cards. However, if, as a result, the total number of points after a widow, will exceed 21 points then a gambler must drop out of a game having shown his cards.

Special rules were established with regard to stakes. Initially, upper and lower bounds were set, and every casino player had a right of choice of a specific stake (within these bounds) depending on the evaluation of his position. If, as a result, it turned out that in accordance with the casino game rules a casino's visitor had a "better" number of points on hand than a casino croupier had, he received a gain in the amount of the stake that he had made, otherwise, this casino player lost his stake. In case of an equal number of points of a gambler and a casino croupier, the game ended in peace, that is the result of the game is considered "harmless" both for a player and a casino.

Let's point out that unlike ordinary casino players a croupier is not obliged to open his cards in that case if the number of points in these cards exceeds 21. Moreover, after all the casino players have opened their cards, and therefore, all the stakes go to a casino gamblers cannot practically find out what was the number of points of a casino croupier, in order to build their game strategy for the next game (whether to risk or not to stand pat, and etc). It goes without saying, it gives a casino croupier considerable advantages. Besides, all the casino players are surely aware of this, and, continue to play. Nothing can be done about it, who does not take risks, as is known, does not win.

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