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Morendil comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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Sorry, I meant to say I'd urge you to press on with the formalization and calculation in your interpretation of the Delta case.
I'll punt on the wall-clock idea. I'm not planning to spend any time working out the formalization for anything that involves large numbers of values for any given variable - my skills aren't up to doing that confidently, and we seem to have enough to go on with formulations of the problem that only involve smaller sets.
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.