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The key question: what kind of dystopian society is this that doesn't have calendar watches?

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Robin's comparison to the assistant is a valid explanation for why the probability is 1/3. But in my opinion, it fails to convince because it doesn't isolate the reason why 1/2 is wrong.

The root cause of the controversy is confusing the occurrence of an outcome with the observation of that outcome. Heads and Tuesday is *not* excluded from occurring, it is just not observed by Sleeping Beauty when it does. The event itself still happens. A random time during this experiment has a 1/4 chance to be any of the four combinations Heads and Monday, Tails and Monday, Heads and Tuesday, or Tails and Tuesday. What Sleeping Beauty knows, that changes these probabilities, is that she won't observe Heads and Tuesday, not that it won't happen.

The answer to the question is now quite simple:

Pr(Heads|Observe) = Pr(Heads and Observe)/Pr(Observe) = (1/4)/(3/4) = 1/3.

Here's a better version of the Assistant, that makes this clearer. I thought it up independently, and found this site when researching where I should go with it:

Rip van Winkle is given a bedroom on the other side of the building from Sleeping Beauty's. He is never given the stay-asleep drug, but is given the amnesia drug Monday night. He has no contact with Sleeping Beauty, and no way to tell what day it is. But he gets asked the same question, about the same coin flip, on Monday. On Tuesday, he is asked the question again if the flip was tails. If it was heads, he is released from the experiment without being asked the question.

The only difference in the problems, is that where Sleeping Beauty cannot observe a time on Tuesday at all, Rip van Winkle can observe it and distinguish it from the other three possibilities. But if he observes that the question is asked, Rip's information is identical to Beauty's. And his answer is trivially 1/3, by the method I used above.

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