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AlephNeil comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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OK but intuitively it can't make any difference whether SB is woken at a fixed or a random time of day, and it can't make any difference whether there is a clock on the wall.
So the solution to the 'random-waking, clock on wall variation' must be the same as the solution of the original SB problem.
See this for a crisp, simple formalization which appears to show where the ambiguity between 1/2 and 1/3 comes from.