philosophies Article To Be or to Have Been Lucky, That Is the Question Antony Lesage 1 and Jean-Marc Victor 2, *   Citation: Lesage, A.; Victor, J.-M. To Be or to Have Been Lucky, That Is the Question. Philosophies 2021, 6, 57. https://doi.org/10.3390/ philosophies6030057 Academic Editors: Fabien Paillusson and Marcin J. Schroeder Received: 21 April 2021 Accepted: 6 July 2021 Published: 9 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Physico-Chimie des Électrolytes et Nano-Systèmes Interfaciaux, PHENIX, Sorbonne Université, CNRS, F-75005 Paris, France; antony.lesage@sorbonne-universite.fr 2 Physique Théorique de la Matière Condensée, LPTMC, Sorbonne Université, CNRS, F-75005 Paris, France * Correspondence: victor@lptmc.jussieu.fr Abstract: Is it possible to measure the dispersion of ex ante chances (i.e., chances “before the event”) among people, be it gambling, health, or social opportunities? We explore this question and provide some tools, including a statistical test, to evidence the actual dispersion of ex ante chances in various areas, with a focus on chronic diseases. Using the principle of maximum entropy, we derive the distribution of the risk of becoming ill in the global population as well as in the population of affected people. We find that affected people are either at very low risk, like the overwhelming majority of the population, but still were unlucky to become ill, or are at extremely high risk and were bound to become ill. Keywords: ex ante chances; dispersion of chances; chronic diseases; gambling; statistical test; twin studies; principle of maximum entropy 1. Introduction “That evening he was lucky”: what do we mean by this? It is even weirder when we say: “the luck turned”. Does this mean that we could be visited by fortune? Or that some people are luckier than others on certain days? Of course, we cannot rule out the fact that some people may bias the chances of success simply by cheating. Yet, is there any way to assess the dispersion of chances among gamblers (or just the fraction of cheaters)? This kind of question is part of the field of probability calculus, which aims at deter- mining the relative likelihoods of events (for a nice historical introduction to probability theory, see [1]). Probability calculus started during the summer of 1654 with the correspon- dence between Pascal and Fermat precisely on the elementary problems of gambling [2]. Symmetry arguments are at the heart of this calculus: for example, for an unbiased coin, the two results—heads or tails—are a priori equivalent and therefore, have the same proba- bility of occurrence of 1/2. This is why it is not anecdotal that Pascal wanted to give his treatise the “astonishing” title “Geometry of Chance”. Another illustration of the power of symmetry arguments is the tour de force of Maxwell who managed to calculate the velocity distribution of particles in idealized gases [3]. At the time when he derived what is since called the Maxwell–Boltzmann distribution, there was no possibility to measure this distribution. It was almost 60 years before Otto Stern could achieve the first experimental verification of this distribution [4], around the same time when he confirmed with Walther Gerlach the existence of the electron spin [5], for which he won the Nobel Prize in 1944. The agreement between theoretical and experimental distributions was surprisingly good. Since its invention in the middle of the 17th century, probability calculus has accompanied most if not all new fields of science, especially since the beginning of the 20th century with the burst of genetics and quantum physics up to the most recent developments of quantum cognition [6], not to mention its countless applications in finance and economy. In probability theory, events are usually associated with random variables that are measurable. For example, in the heads or tails game, heads may be associated with 1 and tails with 0. Then, for a given number N of draws, one can count the number of times the Philosophies 2021, 6, 57. https://doi.org/10.3390/philosophies6030057 https://www.mdpi.com/journal/philosophies