A statistical analysis of luck


A modal analysis of luck, due to Duncan Pritchard, has become quite popular in recent years. There are many reasons to like Pritchard’s analysis, but at least two compelling problems have been identified. So I propose an alternative analysis of luck based on the laws of statistical mechanics. The statistical analysis avoids the two problems facing Pritchard’s analysis, and it has many other attractive features.

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  1. 1.

    Events which, colloquially, we are apt to call unlucky—perhaps because they are in some way negative for someone—count as ‘lucky’ in my sense of the term. And I use the term ‘non-lucky’ for events which feature no luck at all.

  2. 2.

    The PAL is the most recent of Pritchard’s analyses of luck, which is why I focus on it. See (Pritchard 2005) for another influential analysis. The same sorts of problems, it turns out, arise for both.

  3. 3.

    This clarification is important because condition 2 should not be satisfied if the agent fails to form the belief that b at all. For if condition 2 were satisfied whenever the agent failed to form the belief that b, then the PAL would classify many intuitively non-lucky events as lucky. For instance, suppose Millie looks at a working clock and forms the true belief that it is 6:00 a.m. There are many close possible worlds in which Millie fails to form this belief, simply because in those worlds, it is a little earlier or a little later than 6:00 a.m. So on this other reading of condition 2, the PAL incorrectly classifies the event of Millie coming to believe that it is 6:00 a.m. as lucky. See Pritchard (2005, p. 146) for more discussion.

  4. 4.

    For examples of the widespread view that luck and knowledge are incompatible, see Zagzebski (1996).

  5. 5.

    To avoid complications relating to necessary propositions, Pritchard requires that p be contingent.

  6. 6.

    Anti-luck safety conditions on knowledge are not uncontroversial, however. See Sosa (2015, pp. 118–119) for a counterexample to one of Pritchard’s anti-luck conditions.

  7. 7.

    I have changed some of the inessential details of the case.

  8. 8.

    For a detailed discussion of beliefs formed about lottery propositions, see Hawthorne (2004).

  9. 9.

    The same problem arises for other accounts of luck due to Pritchard, such as the one in his (2005, p. 128). That account, which is very similar to the PAL, invokes the imprecise notions of ‘sufficiently large’ classes of worlds and ‘relevant’ initial conditions. So McEvoy’s counterexample applies to it too.

  10. 10.

    The points are represented by 6n-tuples, where n is the number of particles in the universe. The state of each particle is represented by six coordinates: one for its position and one for its momentum along each of the x-axis, the y-axis, and the z-axis. For every assignment of positions and momenta to all n particles, there is a corresponding phase space point, and thus a corresponding possible world.

  11. 11.

    To keep things simple, I have defined macrostates and microstates in terms of phase space, which is principally used in classical physical theories. Additional technical machinery would need to be brought to bear if one wanted to assume that the fundamental dynamical laws are quantum mechanical rather than classical. For example, one of the laws I discuss later (PH) would have to be reformulated, because in quantum mechanics, entropy is defined using Hilbert space rather than phase space.

  12. 12.

    Given the simplifying assumption that the dynamics are deterministic, each microstate lies on exactly one of these trajectories.

  13. 13.

    \(R_{M^{\prime }}\) is the region of phase space corresponding to macrostate \(M^{\prime }\).

  14. 14.

    See Albert (2000) for a detailed discussion of PH.

  15. 15.

    Compatibility with \(R_{PH}\) is necessary to avoid a problem known as the reversibility objection. See Albert (2000, pp. 71–78) for details.

  16. 16.

    More rigorously: according to SP, the right probability distribution is the distribution Pr that is uniform—according to the standard Lebesgue measure—over those regions of phase space that are compatible with PH and with M.

  17. 17.

    It does not matter whether M occurs in the past, present, or future.

  18. 18.

    If the fundamental laws of the universe are indeterministic, one can extend the SAL to account for indeterministic microscopic events: just incorporate the probabilities invoked in the fundamental laws into the SAL in the right way. The details of that, of course, would depend on which indeterministic laws are at issue.

  19. 19.

    As Loewer (2012, p. 129) observes, this is perhaps most plausible on the Best System account of lawhood.

  20. 20.

    See Pritchard (2014, p. 611) for Pritchard’s response to Lackey’s claims.

  21. 21.

    Pritchard (2005, p. 126) raises an additional objection: probabilistic analyses classify some events as lucky, even if those events are not significant for anyone. Years later, Pritchard himself argued that analyses of luck should not include any such ‘significance condition’: even though we may not attribute luck to them, he claimed, such events may in fact be lucky (2014, p. 604). So proponents of the SAL can respond to this third objection in the manner Pritchard eventually did. There is also another option: if you think the SAL requires some sort of significance condition in order to adequately characterize luck, just add one. For example, simply stipulate that, to be lucky, an event must also be significant to at least one agent.

  22. 22.

    This is part of the reason why, in principle, the Mentaculus may be able to support other special science explanations. See Albert (2015, p. 17) for a description of exactly how that would go.

  23. 23.

    Thanks to an anonymous reviewer for raising these issues.

  24. 24.

    As argued by Ulrike Hahn and Paul Warren (2009), there may be a rational basis for believing that irregular sequences are less likely than regular sequences. People often track local subsequence frequencies, not global frequencies, when making probability judgments (Hahn and Warren 2009, pp. 455–456). Perhaps subjects made luck attributions on the basis of these other events—the events of landing on these particular sectors—because subjects were focused principally on local rather than global probabilities.

  25. 25.

    Thanks to an anonymous reviewer for this observation.

  26. 26.

    Or ml(E) is infinite: the range of the ml function is not explicitly specified in the account.

  27. 27.

    Or at least, ml(E) should be normalizable to \(\frac{1}{2}\). And at the very least, ml(E) should be defined.


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Thanks to David Black, Laura Callahan, Sam Carter, Eddy Chen, Jill North, Duncan Pritchard, Jonathan Schaffer, Ernie Sosa, Dean Zimmerman, the audience in Harry Crane‘s “Foundations of Probability” seminar, two anonymous referees, and especially David Albert and Barry Loewer for many helpful comments. This paper was made possible by the generous support of a Society of Christian Philosophers’ GSCT grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation.

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Correspondence to Isaac Wilhelm.

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Wilhelm, I. A statistical analysis of luck. Synthese 197, 867–885 (2020). https://doi.org/10.1007/s11229-018-1745-4

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  • Luck
  • Epistemology
  • Probability
  • Philosophy of science
  • Statistical mechanics