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 game of twenty-one. But let’s not ride before the hounds.
In accordance with the game rules of twenty-one of that time one croupier dealt gamblers two cards each out of a thoroughly shuffled pack consisting of 52 cards. Gamblers themselves did not show their cards to a dealing croupier. At the same time out of two cards taken for him an official of a casino showed one of them (usually the first one) to gamblers. Gamblers 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, nine took 9, and etc). That gambler 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 gambler (including a 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 gambler 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 game rules a casino’s visitor had a “better” number of points on hand than a croupier had, he received a gain in the amount of the stake that he had made, otherwise, this gambler lost his stake. In case of an equal number of points of a gambler and a croupier, the game ended in peace, that is the result of the game is considered “harmless” both for a gambler and a casino.
Let’s point out that unlike ordinary gamblers 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 gamblers 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 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 croupier considerable advantages. Besides, all the gamblers 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.
Thorp managed to find out that owners of gambling houses gave their officials rather strict directions with regard to the strategies which they should stick to in the game with visitors. Control over fulfillment of these directions had its initial aim to prevent from a frame-up of a croupier with the rest of the gamblers, a chance of which could not be excluded. Assigned for a croupier strict rules determining his game strategy really substantially reduced a probability of such a frame-up, but on the other hand, allowed an “advanced” gambler to rather adequately reveal the essence of this strategy and effectively oppose it. For unlike a croupier a gambler needn’t show the first of the received cards, as well as isn’t enchained by any strict rules as regards his strategy, that is why flexibly changing his behavior he can confuse a croupier. For example, Thorp found out that practically in all gambling houses of Nevada State croupiers were strictly ordered to keep away from a widow in case the amount of points in his cards exceeded or was equal to 17, and a player, from our mathematician’s point of view did not have to miss an opportunity to make use of the knowledge of even some aspects of a croupier’s strategy for achievement of his aims. Thus, those advantages which had an official of a gambling house from the start (as we already know, he is not obliged to open his cards at the end of the game), can be compensated to a certain degree for the knowledge of a player about the strategic “tunnel vision” of a croupier.
Besides, as has been mentioned, Thorp, while building his strategy presumed that cards were not often shuffled, in particular, if after finishing of a regular game there were still cards left in a pack, a croupier did not collect the thrown-away by the gamblers cards but dealt them anew (and the next game was played), and only after complete exhaustion of a pack, an official of a gambling house collected all the cards, thoroughly shuffled them and a new “cycle” began. Naturally, if a gambler had a good memory he could change his strategy depending on the knowledge of the cards which had gone out of the game, and what cards could still be counted upon. It is important to remember that a croupier himself who was to strictly follow the directions of the casino’s owners practically without changing his strategy!
Thorp set himself a task to formulate the rules which would allow him to calculate probabilities of taking out one or another card out of an incomplete pack. Knowing these probabilities a gambler could already with reasonable assurance draw cards from the widow without being too much afraid of “a pip out”, and besides, on the basis of the knowledge of some aspects of a croupier’s strategy to make suppositions about those cards which he had, and other gamblers as well. Naturally, as a gambler was to make a decision with regard to a widow very quickly, the sought rules for calculation of probabilities were to be rather simple for a gambler to be able to use them “in mind” with the help of neither a calculator, nor a pen and paper (even if we suppose that a gambler will be given a chance to do calculation on paper, it will certainly arise suspicion). Edward Thorp managed to solve this mathematical problem having created rather simple algorithms for calculation of probabilities of taking out of one or another card from a pack, and using them to build a strategy of the game of twenty-one which would not be very complicated, allowing a gambler to considerably increase his chances of winning!
As the Hungarian mathematician A.Reni states after a few days of presenting his report on the obtained results at the meeting of the American Maths Society in 1960 in Washington “Thorp received from a businessman a letter with a check for 1 thousand dollars intended for checking of a winning strategy in practice. Thorp accepted the check and having learnt the formulated by him rules left for Nevada to try his discovery. The trial went well: less than after two hours Thorp won 17 thousand dollars.
Needless to say, the owner of a gambling house didn’t share Thorp and his companion’s delight with regard to a successful come out of the trial and the next day did his best to prevent Thorp from joining in the game. Later on Thorp tried to penetrate into other gambling houses, but the news of him had already spread far and wide, so that the doors of all the gambling houses appeared to be closed for him. Several times having adjusted a fake beard or having got a makeup of a Chinese, Thorp managed to get to the gaming-table, but in any disguise his constant gain invariably gave him away. Thorp had to refuse from further checking of the strategy developed by him”. Though “additional checks” were “necessary” only to enrich the pockets of the talented mathematician. One could hardly doubt that Either managed to create a real winning strategy!
However, since he could no longer benefit from his discovery himself, he decided to render “welfare assistance” to his colleagues having published in 1961 a small article in an American academic journal (Thorp E.O. “A favorable strategy for twenty-one”, Proc.Nat.Acad.Sci., 47, 110-112, (1961)). And despite the small size of the article and, consequently, an extremely condensed form of persentment, made it comprehensible for rather a narrow group of professionals, one can be sure that a number of American scientists and their friends certainly “improved” their material situation (owners of gambling houses were unlikely to read scientific magazines at that time).
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, cannot get more 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.