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The Shooting-Room Paradox and Conditionalizing on Measurably Challenged Sets

Bartha, Paul and Hitchcock, Christopher (1999) The Shooting-Room Paradox and Conditionalizing on Measurably Challenged Sets. Synthese, 118 (3). pp. 403-437. ISSN 0039-7857. doi:10.1023/A:1005100407551.

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We provide a solution to the well-known “Shooting-Room” paradox, developed by John Leslie in connection with his Doomsday Argument. In the “Shooting-Room” paradox, the death of an individual is contingent upon an event that has a 1/36 chance of occurring, yet the relative frequency of death in the relevant population is 0.9. There are two intuitively plausible arguments, one concluding that the appropriate subjective probability of death is 1/36, the other that this probability is 0.9. How are these two values to be reconciled? We show that only the first argument is valid for a standard, countably additive probability distribution. However, both lines of reasoning are legitimate if probabilities are non-standard. The subjective probability of death rises from 1/36 to 0.9 by conditionalizing on an event that is not measurable, or whose probability is zero. Thus we can sometimes meaningfully ascribe conditional probabilities even when the event conditionalized upon is not of positive finite (or even infinitesimal) measure.

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Additional Information:© 1999 Kluwer Academic Publishers. We appreciate the help of Richard Johns, who first thought of the truncated finite game (of Section 3.1), and the assistance of Ed Perkins for suggestions on how to set up the outcome space in Section 4.1. We also thank all those who have huddled with us around tables in dark corners of hotel bars, discussing this paradox (especially Jamie Dreier, Alan Hájek and Luc Bovens). Finally, we are grateful to two anonymous referees for helpful suggestions.
Issue or Number:3
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:44758
Deposited On:09 Apr 2014 15:04
Last Modified:10 Nov 2021 16:56

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