Solving the Hard Problem of Bertrand's Paradox

Diederik Aerts, Massimiliano Sassoli de Bianchi

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)

Abstract

Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy" problem and a "hard" problem. The easy problem can be solved by formulating Bertrand's question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace's principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible "ways of selecting" an interaction between the entity under investigation and that producing the randomization. This consists in evaluating a huge average over all possible "ways of selecting" an interaction, which we call a 'universal average'. Following a strategy similar to that used in the definition of the Wiener measure, we calculate such universal average and therefore solve the hard problem of Bertrand's paradox. The link between Bertrand's problem of probability theory and the measurement problem of quantum mechanics is also briefly discussed.
Original languageEnglish
Article number083503
Pages (from-to)1-17
JournalJournal of Mathematical Physics
Volume55
Issue number8
DOIs
Publication statusPublished - 2014

Bibliographical note

15 pages

Keywords

  • physics.hist-ph
  • math-ph
  • math.MP
  • math.PR
  • quant-ph

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