The emergence of probability theory

Throughout the Christian Middle Ages, events were believed to follow "divine will." Predicting them through calculation was therefore wholly alien to learned thought. Jérôme Cardan is commonly regarded as the first to address the subject, in his book De ludo aleae, written in 1564. His role should not be overstated: the Italian physician merely intended to write a book explaining how to win at dice. It nevertheless contains the beginnings of the concept of expected value. Drawing on Aristotle, Cardan sought fairness in games: for him, the stakes put up by players A and B should each be proportional to that player's probability of winning.

Let p denote player A's probability of winning (and hence 1 – p that of B), and let m_A and m_B denote their respective stakes. This gives ${\displaystyle \frac{m_{\text{A}}}{m_{\text{B}}}}={\displaystyle \frac{p}{1-p}},\backslash dfrac\{m\_\{\backslash text\{A\}\}\}\{m\_\{\backslash text\{B\}\}\}\; =\; \backslash dfrac\{p\}\{1-p\},$ or pm_B – (1 – p) m_A = 0, which requires each player's expected value to be zero. The text, however, remained little known, since it was not published until a century after it was written.

At the beginning of the 15^th century, Cosimo de' Medici, the future Duke of Tuscany, asked his celebrated tutor Galileo why, when three dice are rolled, a total of 10 appears more often than a total of 9, even though each can be obtained in six ways. The young man must have rolled the dice a great many times, since the respective probabilities are 27/216 and 25/216, a difference of less than 1%. Until then, however, discussion of such problems had remained confined to dice games and had not led to broader mathematical ideas or theories.