Evelyn Marie Adams won the New Jersey state lottery twice in the span of five months. The New York Times described this event as a “1 in 17 trillion” long shot, referring to the probability that one would win twice if one bought one ticket each for exactly two New Jersey state lotteries. Others pointed out that because Adams actually bought many tickets to many different New Jersey state lotteries, the probability that she would be a double winner is somewhat higher than this number. It is, nonetheless, very small. A likelihoodist would be forced to say that her double win favors to an enormously high degree the hypothesis that the lottery was somehow rigged in her favor over the hypothesis that the lottery was fair.

In his (2012), Elliott Sober writes that the Law of Likelihood “seems to endorse the naïve impulse to see conspiracies everywhere, to always think that a hypothesis of Causal Connection is better than the hypothesis of Mere Coincidence.” After all, an apparent coincidence typically involves a co-occurrence of two events that would be much more probable if the events were causally connected than if they occurred independently. Thus, the likelihood ratio of a suitable hypothesis of Causal Connection to a hypothesis of Mere Coincidence is typically very large in cases of coincidence such as the Adams case.

A common reply to the claim that a marvelous coincidence such as Adams’s double win indicates some kind of Causal Connection is that one must put that coincidence into proper perspective. In the case of Adams’s double win, one should consider not the probability that Adams would win the New Jersey lottery twice, but that somebody, somewhere would win some state lottery twice. The probability of the latter is in fact very high (Diaconis and Mosteller 1989), so when applied correctly the Law of Likelihood implies, as it should, that Adams’s double win does not indicate any kind of conspiracy.

This reply has great intuitive appeal, but it seems to violate one of the precious few claims on which philosophers nearly unanimously agree: the Principle of Total Evidence. This principle says, roughly, that one should as far as one can take into account all of the relevant information one possesses in evaluating the truth of some claim. Discarding the information that it was Adams specifically who won the New Jersey lottery specifically violates the Principle of Total Evidence. A likelihoodist cannot claim that it is permissible to discard this information because it is irrelevant: if it were irrelevant, then discarding it would not make any difference, but the only reason a likelihoodist would want to discard it is that doing so does in fact make an enormous difference.

Sober does not claim that the Adams case is a genuine counterexample to the Law of Likelihood, but he does say that it “seems unable to identify a respect in which the [hypothesis of Mere Coincidence] is superior to the [hypothesis of Causal Connection].” His solution to this perceived difficulty is to turn his attention from questions about the evidential meaning of Adams’s double win to questions about the expected predictive accuracy of complex models that allow for conspiracies relative to simple models that do not.

Sober’s appeal to model selection methods in this context is quite interesting, but my present concern is with the Law of Likelihood. It is possible to defend the Law of Likelihood in this case by pointing out that the fact that Adams’s double win generates an enormous likelihood ratio for a hypothesis of Causal Connection against a hypothesis of Mere Coincidence does not imply that a conspiracy theory is belief-worthy. Likelihoodism is a theory of evidential favoring, not a theory of belief. Moreover, it interfaces very nicely with a Bayesian theory of belief that yields intuitively reasonable results in this case given reasonable prior probabilities as inputs.

This defense of likelihoodism is fine as far as it goes, but it fails to vindicate likelihoodism as a viable alternative to Bayesianism. Moreover, it also fails to vindicate the sense that there is something right about the idea that what is relevant is not the probability of Adams’s double win, but rather the probability that somebody, somewhere would have a double win.

I propose that weakening the data by considering simply the probability that somebody, somewhere would have a double win is warranted because it provides a relatively easy way to evaluate the evidential significance of Adams’s double win not for the hypothesis that Adams herself was involved in some kind of conspiracy, but for broader hypotheses of the “everything happens for a reason” flavor. Stories about coincidences like Adams’s double win are popular not because they might have some relatively mundane explanation (such as that she managed to forge the winning tickets), but because they flatter our tendency toward magical thinking in a way that can be enjoyable even for those who know better.

When the general (and vague) hypothesis that some kind of magical or divine forces are at work is at issue, it makes good sense to consider the probability that somebody, somewhere would have a double win, for Pr(Adams’s double win; mere coincidence)/Pr(Adams’s double win; magical forces) is roughly the same as Pr(somebody’s double win; mere coincidence)/Pr(somebody’s double win; magical forces) as long as there is nothing special about Adams that would lead one to think that any magical forces that might be at work would be likely to favor her in particular. Weakening the data does not violate the Principle of Total Evidence when the general hypothesis of “larger forces” being at work (as opposed to some hypothesis about Adams in particular) is at issue because the fact that it was Adams in particular who had the double win makes no difference as far as that hypothesis is concerned. Thus, the fact that Pr(somebody’s double win; mere coincidence) is close to one is sufficient to show that Adams’s double win does not in fact indicate to any substantial degree that there are larger forces at work. It would be hard to reach this conclusion without weakening the data because Pr(Adams’s double win; magical forces) is rather difficult to specify.

# Citations

Sober, Elliott. “Coincidences and How to Reason about Them.” *EPSA Philosophy of Science: Amsterdam 2009*. Springer Netherlands, 2012. 355-374.

Diaconis, Persi, and Frederick Mosteller. “Methods for studying coincidences.”*Journal of the American Statistical Association* 84.408 (1989): 853-861.

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Mayo says

One would never want to include all aspects of the data if one were keen to generalize. Anyway, I agree with your assessments, and maintain it is captured neatly in terms of the kind of error or mistaken construal of the data that is of central interest here.

Greg Gandenberger says

“One would never want to include all aspects of the data if one were keen to generalize.” Yes. I take it that generalizing would involve articulating a model according to which the sufficiency principle justifies excluding aspects of the data that one regards as irrelevant. Is that what you have in mind?

“[Your assessments are] are captured neatly in terms of the kind of error or mistaken construal of the data that is of central interest here.” Yes, I think an error-statistical approach could handle this case. I take it that part of the story would be that tests involving “gellerized” hypotheses about Adams would have low severity.

Mayo says

Greg:

“Yes. I take it that generalizing would involve articulating a model according to which the sufficiency principle justifies excluding aspects of the data that one regards as irrelevant. Is that what you have in mind?”

No it is far more general than parametric modeling (where sufficiency might arise), but in gaining any kind of knowledge about the world or generalizing, one does not distinguish data sets or observations in all kinds of ways that they could be distinguished. Instead one groups things. That involves deliberately leaving things out. This is just a very trivial point.No statistics involved.

Don’t understand the point about Adams and gellerization.

Greg Gandenberger says

OK, your point about omitting details for the sake of generalizing is well taken.

My point about Adams and gellerization is that an error-statistical treatment of Adams’s double win can account for the fact that Adams’s double win shouldn’t lead us to believe that she is blessed by Heaven (or some such thing) because that post-designated hypothesis has not been severely tested. It also shouldn’t lead us to reject the broader hypothesis that lotteries are fair because, given that there’s nothing special about Adams, the data are not remarkable on that hypothesis.