alternate clickbait title: Veritasium is wrong
the Sleeping Beauty problem proposes a scenario where a secret, fair coin toss determines how many times you are asked, with no memory of any prior interviews, about the outcome of that coin toss:
- If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.
- If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.
whenever Sleeping Beauty is awoken, she's only asked one question:
“What do you believe is the probability that the coin came up heads?”
does she answer 1/3 because there are twice as many days on which she would be asked after a tails result, or 1/2 because the coin toss was fair?
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first, here's a very different question: “Do you think the coin came up heads or tails?”
the difference between these two questions is the foundation of the entire dilemma: the distinction is that you know the probability of a fair coin flip, whereas you don't know the outcome of this coin flip.
if you were expected to answer heads or tails, your limited perspective as Sleeping Beauty would be relevant; you know that you'll be asked twice if the coin came up tails, and so if you wanted to answer incorrectly the fewest number of times, you ought to guess that. but you weren't asked to guess, nor were you asked to consider how you should guess; you were only asked about the probability of the outcome itself.
this is also where the "universe vs. multiverse" and "reality vs. simulation" analogies at the end of Veritasium's video break down: unlike a fair coin toss, we don't have any prior knowledge of how likely any of those events are. the fact that we're trying to assess the nature of our environment from inside it doesn't change the fact that we don't know shit!
here's a helpful diagram of which events are relevant to which questions:
