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OrphanWilde comments on What's wrong with this picture? - Less Wrong Discussion
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I think the actual property you're looking for is something like "probability that Alice would produce this sequence by means other than honest coin-flipping accurately reported". All-H is higher probability than something random-looking because there are more-likely scenarios where Alice has a motive for reporting it falsely.
If Alice were a randomly chosen computer program or something, then what you're asking for would be more or less a Solomonoff prior, which is quite popular around these parts, but real-world Alices probably have different patterns of likely deception.
After a little more research, the issue appears to be related to Bonferroni inequalities (where we treat every single possible sequence, or explanation for an unusual sequence, as a hypothesis, on a data set of one), and although this is far outside my expertise in statistics, I suspect something -like- a Bonferroni correction might resolve the paradox.