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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1160,
author = {Mostafa K. Ardakani and Shaun S. Wulff},
title = {An extended problem to Bertrand's paradox},
journal = {Discussiones Mathematicae Probability and Statistics},
volume = {34},
year = {2014},
pages = {23-34},
zbl = {1326.60009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1160}
}
Mostafa K. Ardakani; Shaun S. Wulff. An extended problem to Bertrand's paradox. Discussiones Mathematicae Probability and Statistics, Tome 34 (2014) pp. 23-34. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1160/
[000] [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.
[001] [2] J.L.F. Bertrand, Calcul des Probabilities (Gauthier-Villars, Paris, 1889).
[002] [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.
[003] [4] M. Gardiner, Mathematical games: problems involving questions of probability and ambiguity, Scientific American 201 (1959) 174-182.
[004] [5] J. Holbrook and S.S. Kim, Bertrand's paradox revisited, Mathematical Intelligencer 22 (2000) 16-19. | Zbl 1052.60501
[005] [6] E.T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, New York, 2003). | Zbl 1045.62001
[006] [7] E.T. Jaynes, Well-posed problem, Foundations of Physics 3 (1973) 477-493.
[007] [8] L. Marinoff, A resolution of Bertrand's paradox, Philosophy of Science 61 (1994) 1-24. doi: 10.1086/289777.
[008] [9] G.J. Szekely, Paradoxes in Probability Theory and Mathematical Statistics (Kluwer Academic Publishers, New York, 1986). | Zbl 0605.60002