When in one outcome one person exists and in the other outcome two people exist it may mean that you are twice as likely to exist on the second outcome (if there is random sampling) and then thirder reasoning you describe is correct. Or it may mean that there are just “two of you” in the second scenario, but there are always at least one of you, and so you are not more likely to exist in the second scenario.
Consider these two probability experiments:
Experiment 1: Brain fissure
You go to sleep in Room 1. A coin is tossed. On Heads nothing interesting happens and you wake up as usual. On Tails you are split into two people: your brain is removed from the body, the two lobes are separated in the middle and then the missing parts are grown back, therefore creating two versions of the same brain. Then these two brains are inserted into perfectly recreated copies of your original body. Then on random one body is assigned to Room 1 and the other is assigned to Room 2. Both bodies are returned to life in such a manner that it’s impossible to notice that something happened.
You wake up. What’s the probability that the coin was Heads? You see that you are in Room 1. What is the probability that the coin was Heads now?
Experiment 2: Embryos and incubator
There are two embryos. A coin is tossed. On Heads one embryo is randomly selected and put into an incubator that grows it into a person who will be put to sleep in a Room 1. On Tails both embryos are incubated and at random one person is put into Room 1 and the other person is put into Room 2.
You wake up as a person who was incubated this way. What is the probability that the coin was Heads? You see that you are in Room 1. What is the probability that the coin was Heads now?
On the same day I posted my original comment I later realized what I said was wrong, and I’ll soon edit it to reflect that.
Regarding your response: I think I have a guess on the important difference you’re referring to. They both seem to be equivalent to an Incubator Sleeping Beauty, but see consideration 2 bellow.
1
I think another useful (at least to me) way of seeing/stating what is happening here is that all of the following sentences are true, in an ISB and your two experiments:
The probability (from an external POV) that the coin was Heads or Tails is 1⁄2.
Each individual “me” (however many there are) will experience the coin being Heads or Tails one half of the time.
If every “me” always predicts Heads, all of my mes will be correct 1⁄3 of the time and wrong 2⁄3 of the time. Each individual me will only be able to notice this if we get together after the experiments to compare notes.
I think this is equivalent to the difference in scoring methods you used in Anthropical Motte and Bailey in two versions of Sleeping Beauty.
2
With the two experiments in your response, the only significant difference I can see is that, in experiment 1, there are two identical copies of me, and in 2, there are two different people. I don’t know if you’re implying that this changes any probabilities, and I’m not sure that it does. What I can say is that experiment 2 is, AFAICT, equivalent to the Doomsday argument in it’s setup: two theories on the amount of people that will come to be, with 1:1 prior odds between them, and the question is “should you update on your existing”. I have more reflection to make before I can give any firm answer here, but I’m inclined toward “no”.
3
I have a feeling that, even though we agree with the final probabilities, we disagree on some of the internal details of how these experiments work. What would you say is the significant difference between the experiments, and does it change the numbers?
When in one outcome one person exists and in the other outcome two people exist it may mean that you are twice as likely to exist on the second outcome (if there is random sampling) and then thirder reasoning you describe is correct. Or it may mean that there are just “two of you” in the second scenario, but there are always at least one of you, and so you are not more likely to exist in the second scenario.
Consider these two probability experiments:
Experiment 1: Brain fissure
You go to sleep in Room 1. A coin is tossed. On Heads nothing interesting happens and you wake up as usual. On Tails you are split into two people: your brain is removed from the body, the two lobes are separated in the middle and then the missing parts are grown back, therefore creating two versions of the same brain. Then these two brains are inserted into perfectly recreated copies of your original body. Then on random one body is assigned to Room 1 and the other is assigned to Room 2. Both bodies are returned to life in such a manner that it’s impossible to notice that something happened.
You wake up. What’s the probability that the coin was Heads? You see that you are in Room 1. What is the probability that the coin was Heads now?
Experiment 2: Embryos and incubator
There are two embryos. A coin is tossed. On Heads one embryo is randomly selected and put into an incubator that grows it into a person who will be put to sleep in a Room 1. On Tails both embryos are incubated and at random one person is put into Room 1 and the other person is put into Room 2.
You wake up as a person who was incubated this way. What is the probability that the coin was Heads? You see that you are in Room 1. What is the probability that the coin was Heads now?
Do you see the important difference between them?
On the same day I posted my original comment I later realized what I said was wrong, and I’ll soon edit it to reflect that.
Regarding your response: I think I have a guess on the important difference you’re referring to. They both seem to be equivalent to an Incubator Sleeping Beauty, but see consideration 2 bellow.
1
I think another useful (at least to me) way of seeing/stating what is happening here is that all of the following sentences are true, in an ISB and your two experiments:
The probability (from an external POV) that the coin was Heads or Tails is 1⁄2.
Each individual “me” (however many there are) will experience the coin being Heads or Tails one half of the time.
If every “me” always predicts Heads, all of my mes will be correct 1⁄3 of the time and wrong 2⁄3 of the time. Each individual me will only be able to notice this if we get together after the experiments to compare notes.
I think this is equivalent to the difference in scoring methods you used in Anthropical Motte and Bailey in two versions of Sleeping Beauty.
2
With the two experiments in your response, the only significant difference I can see is that, in experiment 1, there are two identical copies of me, and in 2, there are two different people. I don’t know if you’re implying that this changes any probabilities, and I’m not sure that it does. What I can say is that experiment 2 is, AFAICT, equivalent to the Doomsday argument in it’s setup: two theories on the amount of people that will come to be, with 1:1 prior odds between them, and the question is “should you update on your existing”. I have more reflection to make before I can give any firm answer here, but I’m inclined toward “no”.
3
I have a feeling that, even though we agree with the final probabilities, we disagree on some of the internal details of how these experiments work. What would you say is the significant difference between the experiments, and does it change the numbers?