Chapter 3 Probability

[L]a théorie des probabilités n’est au fond, que le bon sens réduit au calcul. (Laplace 1825, 275)


There’s been a lot to talk about probability since the exchange of correspondences between Pascal and Fermat in 1654, considered “the first substantial work in the mathematics of probability” by (Diaconis and Skyrms 2018, 5). According to Pierre-Simon Laplace, ‘probability theory is basically common sense reduced to calculated.’ In this material, the axiomatic, frequentist and subjective notions of probability, described in detail in Section 2.2 of (S. J. Press 2003) will be used.

Example 3.1 (Frequentist probability) A coin is tossed 100 times approximately under the same conditions and the frequency of heads and tails is observed. If at the end of \(n = 100\) flips we see \(54\) heads and \(100-54=46\) tails, it is calculated that there is \(54/100 = 54\%\) probability of head and \(46/100 = 46\%\) tail.

Example 3.2 (Axiomatic probability) It is assumed that the probability of getting heads on a coin respects the Kolmogorov axioms, described in Section 3.3.5.

Example 3.3 (Subjective probability) When asked about the day of the week on which Christopher Columbus’ maternal grandmother was born, does a person consider it reasonable to attribute – in view of his ignorance to regarding such a question – probability 1/7 for each of the seven days of the week.


Diaconis, Persi, and Brian Skyrms. 2018. Ten Great Ideas about Chance. Princeton University Press.
———. 1974. “Theory of Probability. A Critical Introductory Treatment, 2 Volumes. Translation by a. Machi and a.f.m. Smith of 1970 Book.” Wiley ISBN.
Laplace, Pierre-Simon. 1825. “Essai Philosophique Sur Les Probabilités (1814).” Printed as a Preface to Théorie Analytique Des Probabilités in the Oeuvres Complètes Edition.
Press, S James. 2003. Subjective and Objective Bayesian Statistics: Principles, Models, and Applications, 2nd. Edition. John Wiley & Sons.