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Christiaan Huygens () gave a comprehensive treatment of the subject. From Games, Gods and Gambling ISBN by F. N. David.

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Software - MORE

Christiaan Huygens () gave a comprehensive treatment of the subject. From Games, Gods and Gambling ISBN by F. N. David.

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Software - MORE

and. Gambling. The origins and history of probability and statistical ideas from the earliest times to the Newtonian era. F. N. DAVID, promo.prikol-russkie.online (University of London.

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While contemplating a gambling problem posed by Chevalier de Mere in , Blaise Pascal and Pierre de Fermat laid the fundamental groundwork of probability.

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Christiaan Huygens () gave a comprehensive treatment of the subject. From Games, Gods and Gambling ISBN by F. N. David.

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and. Gambling. The origins and history of probability and statistical ideas from the earliest times to the Newtonian era. F. N. DAVID, promo.prikol-russkie.online (University of London.

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promo.prikol-russkie.online: Games, Gods & Gambling: A History of Probability and Statistical Ideas (): David, F. N.: Books.

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and. Gambling. The origins and history of probability and statistical ideas from the earliest times to the Newtonian era. F. N. DAVID, promo.prikol-russkie.online (University of London.

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promo.prikol-russkie.online: Games, Gods & Gambling: A History of Probability and Statistical Ideas (): David, F. N.: Books.

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While contemplating a gambling problem posed by Chevalier de Mere in , Blaise Pascal and Pierre de Fermat laid the fundamental groundwork of probability.

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David's account, based on the literature of the times, the Romans utilized the astragali with "more zeal" than the Greeks David 5. After Thomas Aquinas' contribution to economic theory, money was seen as "an abstract quantity of reference, and not as a medium of exchange dependent on its value as a metal" Schneider This idea encouraged the use of a device which Schneider called "an ingenious juridical bridge", called "triple contract. How do we find x? Paccioli believed that the stake should be divided proportionally to the scored points. Unfortunately, neither of the solutions posed by Cardano nor Tartaglia is correct, as neither had the idea that a fair division of stakes should be proportional to the probability of winning the whole stake by each player Maistrov Nearly another century would pass before a further controversy over the problem of points would rekindle the discussion. Here is a version of the problem of points: Suppose A and B are playing a series of games where the winner of the overall match is the first player to win a total of k games. Notice now how much the second wins from the first in the second game; he wins 2 ducats minus 4x. Maistrov, "If we assume, as is commonly done, that probability theory owes its origin to gambling, it would be necessary to explain why gambling, which had been in existence for six thousand years, did not stimulate the development of probability theory until the seventeenth century, while in that particular century the theory originated on the basis of the same games of chance. Egyptians played games of "Hounds and Jackals," similar to the present day "Snakes and Ladders", with the help of animal bones. Notice that the unknown author here understands that no matter who wins the next game, the winnings must be equal, or that another x must be won from the other. Using Paccioli's example, we get the ratio of [55; ] or []. Surprisingly enough, this year-old document contains a numerically correct solution of the problem, which some later mathematicians were unable to find. The idea that winnings from winning one game to the next would remain equal is such a revolutionary idea that even Blaise Pascal would have trouble reconciling this to himself, years later. The lender concedes to the merchant a certain fraction of his assumed profit as a guarantee that the investment will be repaid even in case the capital invested in the enterprise is totally or partially lost. But they have to stop playing at the point when one of the two players has won two games and the other none. Schneider interprets the manuscript as follows: The text starts with a description of the following situation: "Two men play schacchi [chess] Each of the players stakes one ducat. The second player ends up winning back x , and the first loses 4x The quantities must then be equal after the depicted fourth game played. Paccioli, frequently referred to in a 16th century citation index under the rubric "error di Fra Luca," was a very careless writer Schneider In addition, Paccioli "borrowed freely from various sources, often without giving the slightest credit -" Smith , as textbook writers continue to do down to the present day, with similar pernicious results. Due to circumstances, they cannot finish the game and one side has 50 points, and the other What share of the prize money belongs to each side? I say that he wins in the game one ducat minus 2x , which the first would have won. Thus I expose my money to danger, just as you in turn expose yours, and the risk of one is judged to be as great as that of the other. There will remain to the first then 4x minus 1 ducat. However, according to Florence N. He observes that Paccioli does not take into account the number of games yet to be won by the players Maistrov This is totally abusurd. Maistrov Paccioli's solution is described by L. The irregular shapes of the bones provided an element of uncertainty, essential for an interesting game. Now take 1 ducat minus 2x from the amount that the first had won from the second, that is 2x. Subsequent to his infamous dispute with Cardano on August 10, , Niccolo Tartaglia published the work "Trattato generale di numerie misure," in , in which dealt with the same problem Cardano had addressed. This event would eventually be recognized as the spark that enflamed Fermat and Pascal to resolve the problem of points. In section 20, titled "Error di Fra Luca dal Borgo," Tartaglia made the following remark against Paccioli: His rule seems neither agreeable nor good, since, if one player has, by chance, ten points and the other no points, then following this rule, the player who has the ten points would take all the stakes which obviously does not make sense. Schneider argues that when gambling was "interpreted in terms of risk taking" in the sixteenth century, problems such as the division of stakes became relevant and of practical importance to the merchants of the time Schneider In essence, according to this interpretation, whose relationship with the mathematical record is obscure, the problem of points addressed the economic needs of the time, rather than being arbitrarily chosen by a select group of individuals. This jurisprudential invention operated as follows:. What reassurance can the lender receive to compensate for such a misfortune? The author continues to use this principle: Notice now for the first, the winner of two games, that if the second had won those two games and were to win the third game, he would clearly win the total remaining part of the first's ducat, and if the first were to win this third game he would win 2 ducats minus 4x. Unfortunately we can only speculate as to the exact conditions of their problem of points, since the first letter s of their correspondence are missing Smith, However, two interesting points can be seen from the remaining letters:. The author continues to say: It is clear that if the loser of the two games would win two further games from his friend neither one would have won anything from the other. Schneider At first sight, it would seem that the lender is at a loss in this contract, since if the capital is lost, and no profit is made, the original loan cannot be repaid. First, by simply offering a solution, Paccioli would had put forward a problem that would later be brought to Cardano's attention. Paccioli "was convinced of the existence of a uniquely determined solution to the problem. Larsen Around , a manuscript written by an unknown author studied by Laura Toti Rigateli was discovered to deal with the problem of points. Similar attitudes towards gambling would propagate centuries after, to 16th century Europe, and down to the present day. Schneider refers to the fact that Paccioli makes the fair distribution of stakes correspond to the number of games already one to each player. Now let us suppose that the second begins to win a game from the first. And in this way the second has to proceed against the risk. This unwavering faith in his solution can possible be mirrored by the following quotation from Paccioli, when confronted with an contrary opinion. In this case he is entitle to winnings of 2 ducats minus 4x from the first, and he is to collect from the first as much as the first would have won, because now each has won two games. A component of risk, infused with the popularity of games of chance, sparked an interest with the merchants of the time. Schneider uses the notation i,j to represent a situation in which the first player has won i games, and the second has won j , and writes DS[a,b] to mean a division of the stakes giving a ducats to the first player, and b ducats to the second. First, Pascal gave two methods to find the solution to the problem of points.{/INSERTKEYS}{/PARAGRAPH} The Greek poet Homer recounted that Patroclus, as a small boy, nearly killed his opponent playing a game of "knucklebones". This may have a psychological impact on the author, since it now becomes more prudent to represent an unknown part of the ducat with a unit of reference, or in keeping with the contemporary "cossic" terminology, a cosa. Thus at the outset the position is 0,0 and the appropriate division of stakes, if the game is halted at this point, would be DS[1,1]. The text then continues, and unfortunately falters from the stated principles, and the author breaks off abruptly Schneider It is unfortunate that this solution went unnoticed for nearly 90 years. Now we have to add 1 ducat on either side and we will have on the one side 4x and on the other 3 ducats minus 4x. Contemporaries of the time believed that such proposal was invalid, and that solutions from this method were questionable Schneider However, Brother Luca was convinced that his method was correct, and that he had found a "uniquely determined solution to the problem. David of this primitive toy: The astragalus has little marrow in it and was possibly not worth cracking for the sake of its contents The popularity of astragali is seen in the variety of cultures that employed its use. The question is, to how much of the opponent's ducat is the leading player entitled? The lender gives a merchant a sum of money as an investment in a commercial enterprise conditional upon the additional payment of a certain proportion of the investment in case of a gain made by the debtor. Here is a description by F. And although this point of view does not offer a just division of stakes, Paccioli had contributed significantly to the problem of points in two significant ways. They play for three games, that is to say the player who first wins three games will receive his opponent's ducat as well as his own. If A needs to take m more games to win the match, while B is n short of the necessary k, how should the stakes be divided? {PARAGRAPH}{INSERTKEYS}The popularity of gambling has been documented throughout history, and even prehistory, over thousands of years. The author then states, without explanation: You must understand that by reason he [the winner] must win in the second game as much as he has won in the first game, so that he has won another x and thus he is now entitled to 2x after two games, while the second who lost is entitled in total to his ducat minus 2x. Thus it may seem natural that mathematicians who lived in the sixteenth century would begin to develop a theory of probability because of their frequent contact with games of chance, and indeed many historians view this development as a result of a few interested mathematicians focusing on particular problems to which their attention had been drawn, relating to games of chance. Note the stake now becomes part of 1 unit, rather than a mass of stakes. Schnider Schnider speculates: Paccioli demonstrates that he was inextricably bound to his own economic model Only to those who accept Paccioli's way of sharing the stakes as reasonable does this refutation appear convincing. The second, if he continues to play, will have 2 ducats minus 4x in the game. This work was the oldest known printed source for the treatment of the problem of points Schneider until the discovery related above. We assume now that the second wins [his] second game. Indeed, many mathematicians chose problems that arose specifically with relation to gambling as examples to study in regard to the computation of probabilities. The lender sells the right to the remaining profit for a part of this profit that must be repaid in any case together with the original investment. Secondly, and most important of all, Paccioli advocated the view that a definite and unique solution exists for the problem of points. However, the contest is interrupted before either competitor has achieved that number. However, some historians of mathematics hold the view that other economic and political pressures caused the evolution of probability theory. We now have an equation to determine the unknown x. In ancient times, ankle bones of animals, or astragali , were used by children as dice are used today. In Cardano's "Practica Arithmetica," published in , he questions Pacioli's solution. The reason is this: if the one who had won two games at first had also won the third game he would win from the second all the rest of the ducat, and this is what conversely the second wins from the first, that is, 1 ducat minus 2x. In fact, this zeal would become so overwhelming, that the Romans eventually banned gaming, except at certain seasons David 8. Tartaglia proposes using the difference of the scores already won to solve the problem. In fact, according to L.