Here’s a probability thought experiment which might make Sleeping Beauty more intuitive to some (at least I’m hoping some people think it’s interesting or could encourage some discussion):
Start with a million simulations of brown-eyed people and one of a blue-eyed person. Everyone fully understands the setup but can’t observe their own eye color. At the beginning, the probability that you have blue eyes is essentially zero (1 in 1,000,001). After a minute passes, all but one of the brown-eyed people get deactivated (they lose consciousness or something along those lines). If you find that you’re still alive, you can now update to a 50% chance of being the blue eyed person and a 50% chance of being the only surviving brown eyed person. Now consider a variant where everyone has a backward-flowing memory; they can see what they’ll observe in the future but have no memory of the past. This time, you start out with one blue eyed person and one brown eyed person; there’s a 50% chance you’re the blue eyed person. After a minute, 999,999 instances of brown-eyed people are created. Once that happens, since you can’t remember if you existed before that moment, you now believe that you’re probably brown-eyed. You might argue that, no, even after the 999,999 brown-eyed instances are created, you should still believe in a 50% chance of being blue-eyed, but the two scenarios are identical except for the direction of time. In the first case, the probability update from ~0% to 50% feels obvious, but in the second case, the same update from 50% to ~0% feels weird.
Here’s a probability thought experiment which might make Sleeping Beauty more intuitive to some (at least I’m hoping some people think it’s interesting or could encourage some discussion):
Start with a million simulations of brown-eyed people and one of a blue-eyed person. Everyone fully understands the setup but can’t observe their own eye color. At the beginning, the probability that you have blue eyes is essentially zero (1 in 1,000,001).
After a minute passes, all but one of the brown-eyed people get deactivated (they lose consciousness or something along those lines). If you find that you’re still alive, you can now update to a 50% chance of being the blue eyed person and a 50% chance of being the only surviving brown eyed person.
Now consider a variant where everyone has a backward-flowing memory; they can see what they’ll observe in the future but have no memory of the past. This time, you start out with one blue eyed person and one brown eyed person; there’s a 50% chance you’re the blue eyed person. After a minute, 999,999 instances of brown-eyed people are created. Once that happens, since you can’t remember if you existed before that moment, you now believe that you’re probably brown-eyed.
You might argue that, no, even after the 999,999 brown-eyed instances are created, you should still believe in a 50% chance of being blue-eyed, but the two scenarios are identical except for the direction of time. In the first case, the probability update from ~0% to 50% feels obvious, but in the second case, the same update from 50% to ~0% feels weird.