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search for: home casino games and mathematics. part 3. after one more year thorp published a book (i mentioned it at the beginning of the article) in which he rather in details, in the form comprehensible to any even a slightly literate and sensible person, set the rules of formation of a winning strategy. but the publication of the book did not only cause a quick growth of those willing to enrich themselves at the cost of gambling houses’ owners, as well as allowed the latter ones to understand the main reason of effectiveness of the developed by thorp strategy. first of all, casinos’ owners understood at last that it was necessary to introduce the following obligatory point into the rules of the game: cards are to be thoroughly shuffled after each game! if this rule is rigorously observed, then a winning strategy of thorp cannot be applied, since the calculation of probabilities of extracting one or another card from a pack was based on the knowledge of the fact that some cards would already not appear in the game! but what does it mean to have thoroughly shuffled” cards? usually in gambling houses the process of thoroughly shuffling” presupposes the process when a croupier, one of the gamblers or, that is still oftener seen of late, a special automatic device makes a certain number of more or less monotonous movements with a pack (the number of which varies from 10 to 20-25, as a rule). each of these movements changes the arrangement of cards in a pack. as mathematicians say, as a result of each movement with cards a kind of substitution” is made. but is it really so that as a result of such 10-25 movements a pack is thoroughly shuffled, and in particular, if there are 52 cards in a pack then a probability of the fact that, for instance, an upper card will appear to be a queen will be equal to 1/13? in other words, if we will, thus, for example, shuffle cards 130 times, then the quality of our shuffling will turn out to be more thorough” if the number of times of the queen’s appearance on top out of these 130 times will be closer to 10. strictly mathematically it is possible to prove that in case our movements appear to be exactly similar (monotonous) then such a method of shuffling cards is not satisfactory. at this it is still worse if the so called “order of substitution” is less, i.e. less is the number of these movements (substitutions) after which the cards are located in the same order they were from the start of a pack shuffling. in fact, if this number equals to t, then repeating exactly similar movements any number of times we, for all our wish, can not get more t different positioning of cards in a pack, or, using mathematical terms, not more t different combinations of cards. certainly, in reality, shuffling of cards does not come down to recurrence of the same movements. but even if we assume that a shuffling person (or an automatic device) makes casual movements at which there can appear with a certain probability all possible arrangements of cards in a pack at each single movement, the question of “quality” of such mixing turns out to be far from simple. this question is especially interesting from the practical point of view that the majority of notorious crooked gamblers achieve phenomenal success using the circumstance, that seemingly “careful shuffling” of cards actually is not such! mathematics helps to clear a situation with regard to this issue as well. in the work gambling and probability theory” a.reni presents mathematical calculations allowing him to draw the following practical conclusion: ” if all movements of a shuffling person are casual, so, basically, while shuffling a pack there can be any substitution of cards, and if the number of such movements is large enough, reasonably it is possible to consider a pack “carefully reshuffled”. analyzing these words, it is possible to notice, that, firstly, the conclusion about “quality” of shuffling has an essentially likelihood character (“reasonably”), and, secondly, that the number of movements should be rather large (a.reni prefers not to consider a question of what is understood as “rather a large number”). it is clear, however, that the necessary number at least a sequence higher than those 10-25 movements usually applied in a real game situation. besides, it is not that simple “to test” movements of a shuffling person (let alone the automatic device) for “accidence”! summing it all up, let’s come back to a question which has been the headline of the article. certainly, it would be reckless to think that knowledge of maths can help a gambler work out a winning strategy even in such an easy game like twenty-one. thorp succeeded in doing it only by using imperfection (temporary!) of the then used rules. we can also point out that one shouldn’t expect that maths will be able to provide a gambler at least with a nonlosing strategy. but on the other hand, understanding of mathematical aspects connected with gambling games will undoubtedly help a gambler to avoid the most unprofitable situations, in particular, not to become a victim of fraud as it takes place with the problem of cards shuffling”, for example. apart from that, an impossibility of creation of a winning strategy for all “cases” not in the least prevents a mathematically advanced” gambler to choose whenever possible the best” decision in each particular game situation and within the bounds allowed by “dame fortune” not only to enjoy the very process of the game, as well as its result. casino games and mathematics. part 1. can the knowledge of mathematics help a gambler to win? one can often hear that the best piece of advice given by a mathematician to a lover of gambling games is an assertion which lies in the fact that the best strategy in gambling 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 gambler 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 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 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 by the mathematician. perhaps, thorp was simply an extremely venturesome person, and his pride both of a gambler and an expert-mathematician was hurt. besides, he could suspect a croupier of dishonesty, since, as he had noticed, cards were not shuffled after each 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 cards after each game, so it was difficult to accuse a croupier of anything. anyway, he managed to develop a winning strategy in the 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 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 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 game of twenty one. – new york: blaisdell,1962.) which made owners of gambling houses in the state of nevada essentially change the rules of the
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