An extended problem to Bertrand's paradox

Mostafa K. Ardakani; Shaun S. Wulff

Discussiones Mathematicae Probability and Statistics (2014)

  • Volume: 34, Issue: 1-2, page 23-34
  • ISSN: 1509-9423

Abstract

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Bertrand's paradox is a longstanding problem within the classical interpretation of probability theory. The solutions 1/2, 1/3, and 1/4 were proposed using three different approaches to model the problem. In this article, an extended problem, of which Bertrand's paradox is a special case, is proposed and solved. For the special case, it is shown that the corresponding solution is 1/3. Moreover, the reasons of inconsistency are discussed and a proper modeling approach is determined by careful examination of the probability space.

How to cite

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Mostafa K. Ardakani, and Shaun S. Wulff. "An extended problem to Bertrand's paradox." Discussiones Mathematicae Probability and Statistics 34.1-2 (2014): 23-34. <http://eudml.org/doc/271059>.

@article{MostafaK2014,
abstract = {Bertrand's paradox is a longstanding problem within the classical interpretation of probability theory. The solutions 1/2, 1/3, and 1/4 were proposed using three different approaches to model the problem. In this article, an extended problem, of which Bertrand's paradox is a special case, is proposed and solved. For the special case, it is shown that the corresponding solution is 1/3. Moreover, the reasons of inconsistency are discussed and a proper modeling approach is determined by careful examination of the probability space.},
author = {Mostafa K. Ardakani, Shaun S. Wulff},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {probability space; probability theory; problem modeling; random chords},
language = {eng},
number = {1-2},
pages = {23-34},
title = {An extended problem to Bertrand's paradox},
url = {http://eudml.org/doc/271059},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Mostafa K. Ardakani
AU - Shaun S. Wulff
TI - An extended problem to Bertrand's paradox
JO - Discussiones Mathematicae Probability and Statistics
PY - 2014
VL - 34
IS - 1-2
SP - 23
EP - 34
AB - Bertrand's paradox is a longstanding problem within the classical interpretation of probability theory. The solutions 1/2, 1/3, and 1/4 were proposed using three different approaches to model the problem. In this article, an extended problem, of which Bertrand's paradox is a special case, is proposed and solved. For the special case, it is shown that the corresponding solution is 1/3. Moreover, the reasons of inconsistency are discussed and a proper modeling approach is determined by careful examination of the probability space.
LA - eng
KW - probability space; probability theory; problem modeling; random chords
UR - http://eudml.org/doc/271059
ER -

References

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  1. [1] L. Basano and P. Ottonello, The ambiguity of random choices: probability paradoxes in some physical processes, Amer. J. Phys. 64 (1996) 34-39. doi: 10.1119/1.18289. 
  2. [2] J.L.F. Bertrand, Calcul des Probabilities (Gauthier-Villars, Paris, 1889). 
  3. [3] S.S. Chiu and R.C. Larson, Bertrand's paradox revisited: more lessons about that ambiguous word, random, Journal of Industrial and Systems Engineering 3 (2009) 1-26. 
  4. [4] M. Gardiner, Mathematical games: problems involving questions of probability and ambiguity, Scientific American 201 (1959) 174-182. 
  5. [5] J. Holbrook and S.S. Kim, Bertrand's paradox revisited, Mathematical Intelligencer 22 (2000) 16-19. Zbl1052.60501
  6. [6] E.T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, New York, 2003). Zbl1045.62001
  7. [7] E.T. Jaynes, Well-posed problem, Foundations of Physics 3 (1973) 477-493. 
  8. [8] L. Marinoff, A resolution of Bertrand's paradox, Philosophy of Science 61 (1994) 1-24. doi: 10.1086/289777. 
  9. [9] G.J. Szekely, Paradoxes in Probability Theory and Mathematical Statistics (Kluwer Academic Publishers, New York, 1986). Zbl0605.60002

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