Archive for Cardano

Game Theory

Posted in Bad Statistics, Books, Talks and Reviews, The Universe and Stuff with tags , , , on September 5, 2009 by telescoper

Nowadays gambling is generally looked down on as something shady and disreputable, not to be discussed in polite company, or even to be banned altogether. However, the  formulation of the basic laws of probability was almost exclusively inspired by their potential application to games of chance. Once established, these laws found a much wide range of applications in scientific contexts, including my own field of astronomy. I thought I’d illustrate this connection with a couple of examples. You may think that I’m just trying to make excuses for the fact that I also enjoy the odd bet every now and then!

Gambling in various forms has been around for millennia. Sumerian and Assyrian archaeological sites are littered with examples of a certain type of bone, called the astragalus (or talus bone). This is found just above the heel and its shape (in sheep and deer at any rate) is such that when it is tossed in the air it can land in any one of four possible orientations. It can therefore be used to generate “random” outcomes and is in many ways the forerunner of modern six-sided dice. The astragalus is known to have been used for gambling games as early as 3600 BC.


Unlike modern dice, which appeared around 2000BC, the astragalus is not symmetrical, giving a different probability of it landing in each orientation. It is not thought that there was a mathematical understanding of how to calculate odds in games involving this object or its more symmetrical successors.

Games of chance also appear to have been commonplace in the time of Christ – Roman soldiers are supposed to have drawn lots at the crucifixion, for example – but there is no evidence of any really formalised understanding of the laws of probability at this time.

Playing cards emerged in China sometime during the tenth century BC and were available in western europe by the 14th Century. This is an interesting development because playing cards can be used for games such as contract Bridge which involve a great deal of pure skill as well as an element of randomness. Perhaps it is this aspect that finally got serious intellectuals (i.e. physicists) excited about probability theory.

The first book on probability that I am aware of was by Gerolamo Cardano. His Liber de Ludo Aleae ( Book on Games of Chance) was published in 1663, but it was written more than a century earlier than this date.  Probability theory really got going in 1654 with a famous correspondence between the two famous mathematicians Blaise Pascal and Pierre de Fermat, sparked off by a gambling addict by the name of Antoine Gombaud, who went by the name of the “Chevalier de Méré” (although he wasn’t actually a nobleman of any sort). The Chevalier de Méré had played a lot of dice games in his time and, although he didn’t have a rigorous mathematical theory of how they worked, he nevertheless felt he had an intuitive  “feel” for what was a good bet and what wasn’t. In particular, he had done very well financially by betting at even money that he would roll at least one six in four rolls of a standard die.

It’s quite an easy matter to use the rules of probability to see why he was successful with this game. The odds  that a single roll of a fair die yields a six is 1/6. The probability that it does not yield a six is therefore 5/6. The probability that four independent rolls produce no sixes at all is (the probability that the first roll is not a six) times (the probability that the second roll is not a six) times (the probability that the third roll is not a six) times (the probability that the fourth roll is not a six). Each of the probabilities involved in this multiplication is 5/6, so the result is (5/6)4 which is 625/1296. But this is the probability of losing. The probability of winning is 1-625/1296 = 671/1296=0.5177, significantly higher than 50%. Sinceyou’re more likely to win than lose, it’s a good bet.

So successful had this game been for de Méré that nobody would bet against him any more, and he had to think of another bet to offer. Using his “feel” for the dice, he reckoned that betting on one or more double-six in twenty-four rolls of a pair of dice at even money should also be a winner. Unfortunately for him, he started to lose heavily on this game and in desperation wrote to his friend Pascal to ask why. This set Pascal wondering, and he in turn started a correspondence about it with Fermat.

This strange turn of events led not only to the beginnings of a general formulation of probability theory, but also to the binomial distribution and the beautiful mathematical construction now known as Pascal’s Triangle.

The full story of this is recounted in the fascinating book shown above, but the immediate upshot for de Méré was that he abandoned this particular game.

To see why, just consider each throw of a pair of dice as a single “event”. There are 36 possible events corresponding to six possible outcomes on each of the dice (6×6=36). The probability of getting a double six in such an event is 1/36 because only one of the 36 events corresponds to two sixes. The probability of not getting a double six is therefore 35/36. The probability that a set of 24 independent fair throws of a pair of dice produces no double-sixes at all is therefore 35/36 multiplied by itself 24 times, or (35/36)24. This is 0.5086, which is slightly higher than 50%. The probability that at least one double-six occurs is therefore 1-0.5086, or 0.4914. Our Chevalier has a less than 50% chance of winning, so an even money bet is not a good idea, unless he plans to use this scheme as a tax dodge.

Both Fermat and Pascal had made important contributions to many diverse aspects of scientific thought in addition to pure mathematics, including physics, the first real astronomer to contribute to the development of probability in the context of gambling was Christiaan Huygens, the man who discovered the rings of Saturn in 1655. Two years after his famous astronomical discovery, he published a book called Calculating in Games of Chance, which introduced the concept of expectation. However, the development of the statistical theory underlying  games and gambling came  with the publication in 1713 of Jakob Bernouilli’s wonderful treatise entitled Ars Conjectandi which did a great deal to establish the general mathematical theory of probability and statistics.