Probability Theory

Abstract

It is customary to set the origins of Probability Theory at the 17th century and relate them to combinatorial problems of games of chance. The latter can hardly be considered a serious occupation. However, it is games of chance that led to problems which could not be stated and solved within the framework of the then existing mathematical models, and thereby stimulated the introduction of new concepts, approaches and ideas. These new elements can already be encountered in writings by P. Fermat, D. Pascal, C. Huygens and, in a more developed form and somewhat later, in the works of J. Bernoulli, P.-S. Laplace, C.F. Gauss and others. The abovementioned names undoubtedly decorate the genealogy of Probability Theory which, as we saw, is also related to some extent to the vices of society. Incidentally, as it soon became clear, it is precisely this last circumstance that can make Probability Theory more attractive to the reader. The first text on Probability Theory was Huygens’ treatise De Ratiociniis in Ludo Alea (“On Ratiocination in Dice Games”, 1657). A bit later in 1663 the book Liber de Ludo Aleae (“Book on Games of Chance”) by G. Cardano was published (in fact it was written earlier, in the mid 16th century). The subject of these treatises was the same as in the writings of Fermat and Pascal: dice and card games (problems within the framework of Sect. 1.2 of the present book). As if Huygens foresaw future events, he wrote that if the reader studied the subject closely, he would notice that one was not dealing just with a game here, but rather that the foundations of a very interesting and deep theory were being laid. Huygens’ treatise, which is also known as the first text introducing the concept of mathematical expectation, was later included by J. Bernoulli in his famous book Ars Conjectandi (“The Art of Conjecturing”; published posthumously in 1713). To this book is related the notion of the so-called Bernoulli scheme (see Sect. 1.3), for which Bernoulli gave a cumbersome (cf. our Sect. 5.1) but mathematically faultless proof of the first limit theorem of Probability Theory, the Law of Large Numbers.