A First Course in Probability and Markov Chains

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Probability measures

The correspondence between Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665) on some matters related to card games suggested by the nobleman and gambler Chevalier de Méré is considered to be the birth of probability. The first treatise, De Ratiociniis in ludo aleae, by Christiaan Huygens (1629–1695) was published in 1647. Other publications followed in the eighteenth century: at the beginning of the century, the Ars conjectandi by Jacob Bernoulli (1654–1705) and the Doctrine of Chances by Abraham de Moivre (1667–1754) were published in 1713 and 1718, respectively. Then the Théorie analytique des probabilités by Pierre-Simon Laplace (1749–1827), which appeared in 1812, summarized the knowledge on probability of the whole century.

The definition of probability given by Pascal for games of chance is the so-called classical interpretation: the probability of an event E is the ratio between the number of successes (the cases in which the event E happens) and the number of all possible cases:

Clearly, the definition can be criticized: it cannot define the probability of those events for which we do not know a priori the number of successes and it does not make sense when there are an infinite number of possible cases. Generally speaking, the definition does not apply to a multitude of cases which we would like to speak of in probability terms.

Another ...