Abstract
In a fair, infinite lottery, it is possible to conclude that drawing a number divisible by four is strictly less likely than drawing an even number; and, with apparently equal cogency, that drawing a number divisible by four is equally as likely as drawing an even number.
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Notes
‘Chance’ here is not intended to imply a notion of objective physical chance like those associated with propensity, frequency, or Humean best-systems accounts, nor do we exclude such interpretations by fiat. We adopt ‘chance’ as a convenient word distinct from ‘probability,’ for we do not assume the standard probability axioms.
This is not meant to be a complete analysis of all the implicit background assumptions in the finest possible granularity. It is just enough to set up our paradox.
(1) and (2) are mutually consistent in this relational context, for here we cannot assert that the chance of fours is some specific value X and then infer by label independence that the chance of even is also X. For a specific chance relation, and proof that it satisfies (1) and (2), see Parker, 2020.
The paradox is thus related to Galileo’s Paradox (Parker, 2009) but suggests further that one’s conception of set size bears on what one should expect in this chance experiment.
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Acknowledgement
This work was partly supported by the John Templeton Foundation under grant #61048. (Matthew W. Parker).
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Norton, J.D., Parker, M.W. An Infinite Lottery Paradox. Axiomathes 32 (Suppl 1), 1–6 (2022). https://doi.org/10.1007/s10516-021-09556-5
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DOI: https://doi.org/10.1007/s10516-021-09556-5