Based on remembering the first pressing, there is a 2⁄3 chance that this really happened and a 1⁄3 chance that this is a false memory
Based on remembering the second pressing, there’s a 1⁄3 chance that you are newly created and your memory of the first pressing is false and a 1⁄3 chance that you are the original. If you don’t remember it you know that you must be the first clone
So both memories are used to figure out what happened.
I think what avturchin is getting at is that when you say “there is a 1⁄3 chance your memory is false and a 1⁄3 chance you are the original”, you’re implicitly conditioning only on “being one of the N total clones”, ignoring the extra information “do you remember the last split” which provides a lot of useful information. That is, if each clone fully conditioned on the information available to them, you’d get 0-.5-.5 as subjective probabilities due to your step 2.
If that’s not what you’re going for, it seems like maybe the probability you’re calculating is “probability that, given you’re randomly (uniformly) assigned to be one of the N people, you’re the original”. But then that’s obviously 1/N regardless of memory shenanigans.
If you think this is not what you’re saying, then I’m confused.
Secondly, you’re right about conditioning on the last split. The original and last clone each think that they have a 50% chance of being the original and everyone knows that they aren’t.
Given this, it’s tough making sense of the problem posed in the original post. Maybe the question isn’t asking about the probability of the original knowing that they are the original at the end, but the chance of someone who thinks they might be the original (including those with false memories) turning out to be the original. Of course it is hard to define exactly what time we are asking about since some of these memories are false. It seems like we need to define some kind of virtual time for it to even make sense. But once this is surmounted, it should be 1/n.
Again, I should be clear, this is one part of anthropics where my ideas are less developed. I think this post will have to be edited once I have a more comprehensive theory.
Consider after the second pressing:
Based on remembering the first pressing, there is a 2⁄3 chance that this really happened and a 1⁄3 chance that this is a false memory
Based on remembering the second pressing, there’s a 1⁄3 chance that you are newly created and your memory of the first pressing is false and a 1⁄3 chance that you are the original. If you don’t remember it you know that you must be the first clone
So both memories are used to figure out what happened.
I think what avturchin is getting at is that when you say “there is a 1⁄3 chance your memory is false and a 1⁄3 chance you are the original”, you’re implicitly conditioning only on “being one of the N total clones”, ignoring the extra information “do you remember the last split” which provides a lot of useful information. That is, if each clone fully conditioned on the information available to them, you’d get 0-.5-.5 as subjective probabilities due to your step 2.
If that’s not what you’re going for, it seems like maybe the probability you’re calculating is “probability that, given you’re randomly (uniformly) assigned to be one of the N people, you’re the original”. But then that’s obviously 1/N regardless of memory shenanigans.
If you think this is not what you’re saying, then I’m confused.
Firstly, what’s 0-.5-.5 mean?
Secondly, you’re right about conditioning on the last split. The original and last clone each think that they have a 50% chance of being the original and everyone knows that they aren’t.
Given this, it’s tough making sense of the problem posed in the original post. Maybe the question isn’t asking about the probability of the original knowing that they are the original at the end, but the chance of someone who thinks they might be the original (including those with false memories) turning out to be the original. Of course it is hard to define exactly what time we are asking about since some of these memories are false. It seems like we need to define some kind of virtual time for it to even make sense. But once this is surmounted, it should be 1/n.
Again, I should be clear, this is one part of anthropics where my ideas are less developed. I think this post will have to be edited once I have a more comprehensive theory.