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A Resolution of Bertrand's Paradox

Published online by Cambridge University Press:  01 April 2022

Louis Marinoff
Louis Marinoff*
Affiliation:
Department of Philosophy University of British Columbia
*
‡Send reprint requests to the author, Department of Philosophy, University of British Columbia, 1866 Main Mall E-370, Vancouver, B.C. V6T 1Z1, Canada.
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Abstract

Bertrand's random-chord paradox purports to illustrate the inconsistency of the principle of indifference when applied to problems in which the number of possible cases is infinite. This paper shows that Bertrand's original problem is vaguely posed, but demonstrates that clearly stated variations lead to different, but theoretically and empirically self-consistent solutions. The resolution of the paradox lies in appreciating how different geometric entities, represented by uniformly distributed random variables, give rise to respectively different nonuniform distributions of random chords, and hence to different probabilities. The principle of indifference appears consistently applicable to infinite sets provided that problems can be formulated unambiguously.

Type
Research Article
Information
Philosophy of Science , Volume 61 , Issue 1 , March 1994 , pp. 1 - 24
Copyright
Copyright © 1994 by the Philosophy of Science Association

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Footnotes

†

I would like to express my gratitude to Bill Boos, Roy Douglas, Dorothy Edgington, J. Howard Sobel, and the referee for Philosophy of Science for their insightful commentaries. This paper was originally read in March 1992, at a Colloquium in the Department of Philosophy, University of British Columbia.

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