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Luke_A_Somers comments on Why Bayes? A Wise Ruling - Less Wrong
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Sorry, I have bad memory for details of this sort, only remember the abstract observation, which is recurrent enough that I have a cached phrase to identify and point out such situations ("This doesn't distinguish the alternatives!"). Could make up some examples, but I don't think it's useful for clarification in this case, and it won't provide further evidence for the existence of the issue.
If the woman who lost hadn't been so comprehensively messed up in the head, you would've had an example in the OP. I wonder if there was a similar test more likely to succeed.