Probability, traditionally introduced through experiments involving coin tosses or dice rolls with combinatorial calculations as the fundamental tool, has a captivating thread of development deeply rooted in geometric games. One of these games, known as the "franc-carreau," was examined by Buffon in the 18th century, marking the commencement of an innovative path within the realm of probability. While traditional challenges in gambling often revolved around determining a bettor's advantage in a particular outcome, this branch possessed a distinctiveness that necessitated an approach based on geometry rather than combinatorial calculus. The earliest printed trace of a study on geometric probability emerges from Thomas Bayes' renowned "Essay," posthumously published in 1763. However, the rediscovery of Newton's manuscripts dating back to 1665 has demonstrated that he also grappled with issues of geometric probability, inspired by his reading of Christiaan Huygens' work in 1657, titled "De Ratiociniis in Ludo Aleae." To address such problems, Bayes developed a definition of probability that allowed for dealing with situations involving an infinite number of possible outcomes. This study retraces the history of the earliest approaches to geometric probability, beginning with the well-known example referred to as the "table of Bayes," proceeding through Laplace's exposition of the famous "Buffon's needle problem," and touching upon some of its recent applications in the field of ecology. We also examine the "Bertrand paradox," which raised doubts about the reliability of geometric probability, along with several attempts at resolution, including that of Montessus. Finally, we turn our gaze to the developments of geometric probability in 19th-century Italy, exploring the "meeting problem" proposed by Bordoni in 1854 and the debate between Giulio Vivanti and Ernesto Cesàro regarding the meaning of geometric probability in light of the "Bertrand paradox." This study provides a comprehensive overview of the evolution of geometric probability and its impact on mathematics and the interpretation of probability itself.
La probabilità nasce e viene insegnata partendo da lanci di monete o di dadi avendo come strumento tecnico di riferimento il calcolo combinatorio. Tuttavia un importante filone della probabilità venne sviluppato partendo da giochi di natura geometrica come ad esempio quello del franc-carreau, studiato da Buffon a metà del XVIII secolo. Come per i problemi tradizionali legati al gioco d'azzardo, si era interessati soprattutto a determinare il vantaggio di uno scommettitore su un esito particolare del gioco. Tuttavia data la particolarità del gioco, non era possibile applicare il calcolo combinatorio direttamente, pertanto per risolvere il problema fu stato necessario applicare un approccio diverso ed usare la geometria. La prima volta che compare a stampa lo studio di un problema di probabilità geometrica risale al celebre Essay di Thomas Bayes, pubblicato postumo nel 1763. La pubblicazione dei manoscritti di Newton nella seconda metà del XX secolo ha mostrato come egli avesse risolto un problema di probabilità geometrica già nel 1665, ispirato dalla lettura del De Ratiociniis in Ludo Aleae che Christiaan Huygens aveva pubblicato pochi anni prima, nel 1657. Proprio per risolvere i primi problemi di con la probabilità geometrica Bayes definisce la probabilità in modo da garantirsi il passaggio al caso in cui vi siano infiniti casi possibili. Partendo da questo primo tentativo altri matematici si sono cimentati in vari approcci di risoluzione di problemi affini a quelli affrontati da Bayes. Lo scopo del capitolo primo è raccontare brevemente la storia dei primi approcci alla probabilità geometrica partendo dal celebre esempio della tavola di Bayes, arrivando all'esposizione di Laplace del celebre problema dell'ago di Buffon e accennando anche ad alcune sue recenti applicazioni in ecologia. Passeremo nel capitolo successivo ad analizzare il paradosso di Bertrand che mise in dubbio laffidabilità della probabilità geometrica. Osserveremo infine qualche riflesso della probabilità geometrica nell'Italia del XIX secolo: dal problema dell'incontro proposto da Bordoni nel 1854 al dibattito tra Giulio Vivanti ed Ernesto Cesaro sul significato della probabilità geometrica alla luce del paradosso di Bertrand.
Probabilità geometrica: dall'ago di Buffon al paradosso di Bertrand
FANTAUZZO, MANLIO
2022/2023
Abstract
Probability, traditionally introduced through experiments involving coin tosses or dice rolls with combinatorial calculations as the fundamental tool, has a captivating thread of development deeply rooted in geometric games. One of these games, known as the "franc-carreau," was examined by Buffon in the 18th century, marking the commencement of an innovative path within the realm of probability. While traditional challenges in gambling often revolved around determining a bettor's advantage in a particular outcome, this branch possessed a distinctiveness that necessitated an approach based on geometry rather than combinatorial calculus. The earliest printed trace of a study on geometric probability emerges from Thomas Bayes' renowned "Essay," posthumously published in 1763. However, the rediscovery of Newton's manuscripts dating back to 1665 has demonstrated that he also grappled with issues of geometric probability, inspired by his reading of Christiaan Huygens' work in 1657, titled "De Ratiociniis in Ludo Aleae." To address such problems, Bayes developed a definition of probability that allowed for dealing with situations involving an infinite number of possible outcomes. This study retraces the history of the earliest approaches to geometric probability, beginning with the well-known example referred to as the "table of Bayes," proceeding through Laplace's exposition of the famous "Buffon's needle problem," and touching upon some of its recent applications in the field of ecology. We also examine the "Bertrand paradox," which raised doubts about the reliability of geometric probability, along with several attempts at resolution, including that of Montessus. Finally, we turn our gaze to the developments of geometric probability in 19th-century Italy, exploring the "meeting problem" proposed by Bordoni in 1854 and the debate between Giulio Vivanti and Ernesto Cesàro regarding the meaning of geometric probability in light of the "Bertrand paradox." This study provides a comprehensive overview of the evolution of geometric probability and its impact on mathematics and the interpretation of probability itself.È consentito all'utente scaricare e condividere i documenti disponibili a testo pieno in UNITESI UNIPV nel rispetto della licenza Creative Commons del tipo CC BY NC ND.
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https://hdl.handle.net/20.500.14239/17213