In my last two articles, I discussed the origins of probability theory and how humans took life for granted, believing that the future was not only unknowable, but somehow pre-ordained. In the sixteenth century, through the popularity of games and especially gambling, it appears that we started to ask if we could ordain the future. Could we look at discreet future events and understand the “chances” of something specific happening? The Italian polymath, Gerolamo Cardano, made the first published attempt that used the correct approach in 1539, although his thesis wasn’t published until 1663.
History shows that nothing about probabilities was published until Blaise Pascal and Pierre de Fermat wrote to each other in 1654 about solving the problem of the “unfinished game”, which I covered at length in Part II of this series. It was also likely discussed at the various academic societies that were popular in France at that time. However, there must have been discussion about Cardano’s method in academic circles, and more than likely amongst gamblers, because sometime between 1613 and 1623, Galileo Galilei responded to a question from his patron, the Grand Duke of Tuscany, about the frequency of the totals of 9, 10, 11 and 12 being thrown with three dice.
The question was related to the game of Hazard, a forerunner of craps. Why does it seem that the totals 10 and 11 appear to come up more frequently than 9 or 12, when they all appear to be capable of being “made” with the same number of combinations: six for each? If we look at just the totals of 9 and 10, we can see they can be made for 9 by the following combinations: 621, 531, 522, 441, 432, 333, and for 10: 631, 622, 541, 532, 442, 433.
Galileo drew a table showing all of the combinations possible with three dice (216) and importantly, how many ways each combination could be arranged. For example, 621 can be arranged six ways, 522 only three ways and 333 only one way. Galileo demonstrated that although 9 and 10 have the same number of possible combinations (6), for 9, they can be arranged 25 ways and for 10, 27 ways. Galileo wrote it with such simplicity and clarity that it is thought by mathematical historians that he must have had prior knowledge of how to attack the problem. He knew that it was not just the total number of combinations that was important, but also the number of arrangements of those combinations.
On August 24, 1654, Blaise Pascal wrote to Pierre de Fermat about his solution to the problem of how to split a pot before a game is finished. Pascal explained his solution to a best-of-five coin-spinning series between Player A (heads) and Player B (tails).
Who were these two men? Blaise Pascal was born on June 19, 1623, in Clermont, France, the son of a wealthy tax collector (how times have changed!). His father had odd ideas about education; he not only forbid his son to study mathematics before he was 15, he wouldn’t allow any books on mathematics into the house. In secret at the age of 12, young Blaise started to work on problems of geometry. He proved to be quite a prodigy, so much so that his father gave up his job and moved to Paris so that he could oversee his son’s education.
Pascal developed many ideas that are essential to mathematics today. When he was 15, his paper on conic sections was so advanced that Descartes thought it must have been written by somebody else. Sadly, late in 1654 when he was 31, Pascal gave up mathematics shortly after his carriage was involved in an accident, which, although unhurt, left him dangling over the River Seine. He devoted what was left of his short life to religious philosophy. He died from a stomach ulcer at the age of 39.
Pierre de Fermat was born on August 17, 1601, in Beaumont de Lamagne, France. His father was a wealthy merchant, trading in skins and leather. A lawyer by training and vocation, his interest in mathematics was purely as an amateur; unfortunately, he published no papers. He worked extensively on number theory and is best known for Fermat’s Conjecture: proof that there is no whole-number solution to x2 + y2 = z2, apart from 3, 4 and 5.
He wrote in the margin of a book about the problem, “I have discovered a truly marvellous proof of this, which, however, the margin is not large enough to contain.” No record of his proof has ever been found and it took an enormous effort over three centuries before Andrew Wiles in 1993 developed the proof, using mathematics not available in the time of Fermat. Most doubt Fermat actually developed the proof as he claimed; perhaps he was just playing a joke on the world.
Fermat’s method to resolve the unfinished game was to write out all the possible outcomes of the spins and determine which Player A and Player B would win. He assumed that Player A had two spins correct and Player B had one; therefore, two further spins would be required to declare a winner. The outcomes would be HH, HT, TT, and TH. In three of these outcomes, Player A would win and in one, Player B would win. Because each of the four outcomes had the same chance of occurring, the pot should be split one-quarter to Player B and three-quarters to Player A.
Pascal agreed with him, although he pointed out that the original questioner, M. le Chevalier de Méré, did not. Méré argued that in practise, if heads was spun on the first go, the game would stop; Player A would have reached three correct. Thus, there were only three possible outcomes: H (game over), TH, TT. Player A would win two and Player B one, therefore the pot should be split two-thirds to Player A and one-third Player B.
Fermat was correct, not de Méré. If you plot it on a decision tree, ascribing the odds at each junction on the tree, you will quickly understand that “H (game over)” represents 50% of the outcomes.
Pascal wanted to know what happened if you increased the complexity of the problem. In a game that involved throwing at least one six in eight rolls of a single die, how should you split the pot if by the fourth roll, a six had not been rolled? And, among others, how about a game with three players instead of two (which Pascal got very wrong)? Working through them together, Pascal and Fermat developed a general method that would work for all outcomes of these type of problems and more. Probability theory, which has allowed modern society to manage many forms of risk, was on its way.
Fermat’s and Pascal’s methods worked on problems where the outcomes were known and the probabilities could be assigned. It took two further advances before probabilities could be inferred from the outcomes already experienced, where the probability is unknown. But a discussion of this will have to wait for another day.
