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AshwinV comments on XKCD - Frequentist vs. Bayesians - Less Wrong Discussion
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In my opinion, sort of. Munroe probably left out the reasoning of the Bayesian for comic effect.
But the answer is that the Bayesian would be paying attention to the prior probability that the sun went out. Therefore, he would have concluded that the sun didn't actually go out and that the dice rolled six twice for a completely different reason.