# Probability and Statistics Seminar

### Probability Calculus - the construction of Pacheco D’Amorim in 1914

At the end of the XIXth Century, the classical definition of Probability and its extension to the continuous case were too restrictive and some geometrical applications, based in ingenious interpretations of Bernoulli-Laplace principle of insufficient reason, led to several paradoxes. David Hilbert, in his celebrated address at the International Congress of Mathematicians of 1900, included the axiomatization of Probability in his list of 23 important unsolved problems. Only in 1933 did Kolmogorov lay down a rigorous setup for Probability, inspired by Fréchet’s idea of using Measure Theory. But before this some other efforts to build up a proper axiomatization of Probability deserve to be more widely credited. Among those, the construction of Diogo Pacheco d’Amorim, in his 1914 doctoral thesis, is one of the most interesting. His discussion of a standard model, based on the idea of random choice instead of the concept of probability itself, seems limited, but his final discussion on how to use the law of large numbers and the central limit theorem to have an objective appraisal of whether sampling made by others, or even by a mechanical device, is indistinguishable from a random choice made by one-self, is impressive, since it anticipates the ideas of Monte Carlo by almost 30 years.