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nshepperd comments on Yes, Virginia, You Can Be 99.99% (Or More!) Certain That 53 Is Prime - Less Wrong
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H^200 isn't any less likely under the assumption that the coin is fair, and the person reporting the coin is honest. But! H^200—being a particularly simple sequence—is massively more likely than most other sequences under the alternative assumption that the reporter is a liar, or that the coin is biased.
So being told that the outcome was H^200 is at least a lot of evidence that there's something funny going on, for that reason.
This has nothing to do with simplicity. Any other apriori selected sequence, such as first 200 binary digits of pi, would be just as unlikely. It seems like it is related to simplicity because "non-simple" sequences are usually described in an aggregate way, such as "100 heads and 100 tails" and in fact include a lot of individual sequences, resulting in an aggregate probability much higher than 1/2^200.
Yes, under the hypothesis that the coin is fair and has been flipped fairly all sequences are equally unlikely. But under the hypothesis that someone is lying to us or has been messing with the coin simple sequences are more likely. So (via Bayes) if we hear of a simple sequence we will think it's more likely to have be artificially created than if we hear of a complicated one.