Tottenham Report: Taking Life for Granted – the Development of Probability Theory, Part II By Andrew Tottenham, Managing Director, Tottenham & Co January 6, 2021 at 10:00 pm This post is the second of three looking into the development of probability theory, an important concept for the development of our industry. In the first part, I discussed how until the late 15th century, our forebears thought that the future was in the “lap of the gods” and could not be predicted, despite what might have been obvious clues. Late in the 15th century, people started to speculate about the fairest way to split the pot in a game that was stopped before its natural conclusion — in other words, “the problem of the unfinished game”. The primary method at the time was to look backward; the pot should be split according to the number of rounds each player had already won. Almost half a century later in 1539, the Italian mathematician Gerolamo Cardano suggested that the fairest way was to look forward rather than backward, to predict all the possible outcomes and split the pot according to the proportion of the number of times each would have won, had the game gone to completion. This, in my opinion, was a pivotal moment, the first time someone had thought to look forward and see what might be. Thus, the future was no longer in “the lap of the gods,” but predictable. So who was Gerolamo Cardano and how did he arrive at this method of solving the problem of the unfinished game? Cardano was born in 1501, the (possibly illegitimate) son of Fazio Cardano, a prominent Milanese lawyer. The young Cardano was extremely intelligent and despite his father’s insistence that he study law, he was allowed to go to the University of Pavia to study medicine. A brilliant student, he made a name for himself performing well in the public “disputes” of the time, in which academics argued their positions on various scientific topics in front of a public audience. Cardano’s father died and left him a small income which he needed to supplement. He had had some success at gambling and threw himself headlong into it. Although he neglected his studies, he was able to earn a living from game playing. Gambling at the time encompassed skill games (chess for money), games partially dependent on skill (backgammon), and games of pure chance (dice games like Hazard – an old form of craps) and a new arrival, card games. The chess of the times was a fast and furious form of speed chess and Cardano excelled, so much so that many refused to play with him. The most popular form of card game in Europe in the Middle Ages was “primero”. The rules have been lost, but it appears to be a precursor to modern poker. Cardano wrote several books. One was a cornerstone for algebra, while another, Liber de Ludo Aleae, was a guide for gamblers. According to his memoirs, there were four volumes, the first three dealing with chess and games like backgammon, but somehow these were lost. Only the volume on dice and card games survived. That book dwells on when to play, with whom, and the effects of gambling. Apparently, gambling produces “kindly feelings” and is useful in times of melancholy. But Cardano warns lawyers that they “gamble at a disadvantage”: If they win, they are called gamblers, and if they lose people may think they are no better at being a lawyer than they are a gambler – my lawyer friends take note! It also examines probabilities. Cardano at first started to examine dice rolls. He showed that there were 36 possible combinations of two dice and, importantly, that the “probability” (not the word he used) of a particular combination coming up would be the number of that particular combination out of the total number of combinations divided by the total number of combinations (36). For example, there are two ways to roll a total of three, 2:1 and 1:2. Therefore, the probability of rolling this combination is 2 divided by 36 or 1/18. He further understood that the probabilities of two linked outcomes are the product of their respective probabilities. To roll two dice twice and to get a combination of a 2 and a 1 followed by a combination of a 6 and a 5 is 1/18 multiplied by 1/18 or 1/324. Some of the dice games at the time involved a player bidding that certain numbers would come up on the next roll. Cardano calculated what he called “equality”, how much someone should bid against that bid to be even after a “circuit” — theoretically, all the possibilities coming up once. He correctly showed that there are 11 possibilities of throwing a 1 on either die, an additional 9 ways for a 2 (ignoring 1:2 and 2:1 that was already accounted for) and 7 ways for a 3 (ignoring 1:3, 3:1, 2:3 and 3:2) and these should be summed to calculate the probabilities. The combinations that are ignored are those that have already appeared in the previous totals. This was ground–breaking stuff! Looking forward as to what the possibilities were, then predicting the various outcomes, was the first known instance of risk management. After working out probabilities with three and four dice, he turned his investigations to the game of primero. At one point in the game, usually only two players are left, all the others having dropped out. One player has to declare his hand, while the other has to make a bid and then can exchange cards or declare a “fare al salvare” and split the pot. Cardano, in the first instance, wanted to know how much the person should bid for “equality”, in other words what was the probability that after exchanging cards they could beat the declared hand or in the event of the “fare al salvare” how the pot should be split to be fair. His calculations for the scenarios he detailed were spot on. In fact, all his calculations for dice and primero were accurate; he made a simple mistake calculating odds for backgammon, which he later corrected. Cardano was trying to develop a systematic approach to solving questions of probabilities for these games. He mainly succeeded, but was not credited for it during his life. Sadly, Liber de Ludo Aleae was not published until almost 130 years later. Not until 1654, 115 years later and 9 years before Cardano’s volume was published, did Blaise Pascal and Pierre de Fermat, two brilliant French mathematicians, pick up the problem of the unfinished game. In a series of letters over two years, they developed probability theory further. More on that in my next article.