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PhilGoetz comments on Monty Hall Sleeping Beauty - Less Wrong Discussion
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No; you are asking her two different questions, so it is correct for frequentism to give different answers to the different questions.
Of course. But the two questions are the same outside of anthropic situations; they are two extensions of the underdefined "how often was I right?" Or, if you prefer, the frequentist answer in anthropic situations is dependent on the exact question asked, showing that "anthropic probability" is not a well defined concept.