Abstract

A central riddle plagues the mathematical field. The probability of a single event that is physical and random does not correspond to reality. It can be said the probability of one uncertain result is nonsensical. Adopting Popper's language, we call this: the single case problem.
It has been recognized by a large circle of researchers all along, but it has not been approached in mathematical-analytical terms. Frequentists and subjectivists, whose philosophies are completely influenced by the unrealism of probability, offer the most apparent example case. The former school focuses on the collective that is a material event but non-single. The latter qualifies personal belief that is a single case but unrelated to experimental control. Each school selects a precise argument in order to bypass probability nonsense. It may be said they both circumvent the riddle using “managerial” criteria that are anything but analytical.
This research proves four theorems that analyze the complicated phenomenology of one aleatory case and make evident the connected questions. In fact, a long quarrel divides classical and Bayesian statisticians who seek the best statistical approach. For example, some criticize Bayesianism because it is more complex in calculations, while others blame frequentism, which requires a large amount of data. The theorems suggest how to use each statistics under the best conditions and how to close this dispute.

P. Rocchi, L.Gianfagna (2011) Deux théorèmes et la nature multiple de la probabilité. Comptes Rendus de la Société Canadienne d’Histoire et de Philosophie des Mathématiques (CSHPM/SCHPM), vol. 24, pp. 110-118.
P. Rocchi (2014) Janus-Faced Probability, Springer.
P. Rocchi (2017) Four foundational questions in probability theory and statistics. Physics Essays, vol. 30, no. 3, pp.314-321 https://arxiv.org/abs/1901.03876
P. Rocchi (2020) Sul Significato della Probabilità e la Scelta del Metodo. Book of short Papers SIS 2020, Pollice A., Salvati N., Schirripa Spagnolo F. (eds), Pearson Publisher, pp. 1326-1331