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PhilGoetz comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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On a random waking moment, Monday preceded by heads is equally likely as Monday preceded by tails.
I think you're thinking of a similar problem that we discussed last year, which involves a forgetful driver who is driving past 1 to n intersections, and needs to turn left at at least one of them. That problem is different, because it's asking about the probability of turning left at least once over the course of his drive.
The absent-minded driver is essentially the same problem, but it's easier to analyze because explicit payoff specification prompts you to estimate expected value of possible strategies. In estimating those strategies, we use the same probability model that would say "1/2" in the Beauty problem.