There’s a probability problem that lacks an obvious solution, despite appearing simple at first glance. It’s usually called the *Sleeping Beauty Problem*, but I’m uncomfortable with that formulation, as it strikes me as needlessly sexist: it usually revolves around a young woman who is put to sleep by researchers, awoken and questioned about the result of a coin flip, then given a mild memory-erasing procedure, then put back to sleep, etc. Maybe it’s not (always) sexist. In some versions, Sleeping Beauty is said to have consented. But anyone can be made to consent to anything in fiction. At any rate, there’s no harm in changing it, and in some ways doing so makes it easier to think about (in other ways, not so much).

What you see here is my attempt at thinking through this difficult problem, which I reformulate as the *Amnesiac’s Dilemma*. (Though I’ll refer to it as the *Sleeping Beauty Problem *as well; I just won’t use that story… much.) The upshot of the dilemma is that 1/2 and 1/3 both seem to be viable solutions (proponents of which have been called halfers and thirders, respectively). Rather than rule the problem indeterminate, we take it that there must be some fact of the matter about which solution is correct given that the problem can be reasonably well-defined by a discrete, finite sample space in which the experiment may be repeated indefinitely.

To make sense of the problem, we need to be clear about its relevant features—for example, about what counts as a desired outcome. It may be that 1/2 is valid for one sort of outcome, while 1/3 is for another. In fact, I ultimately conclude here that, whenever asked “Heads or Tails?”, the Amnesiac (Henry, in this case) should have a credence of 1/2 that the coin landed Heads, even though he may (reasonably) simultaneously assign 1/3 credence to his situation being, say, {(First Question AND Heads)} (Henry’s analogue to Sleeping Beauty’s {(Monday AND Heads)}). Though I don’t take this result lightly (if pushed, I might lean towards 1/2 or agnosticism). This will make more sense (I hope!) by the end.

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