> However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails.
Beauty can't lawfully reason about the problem, while treating awakenings as individual outcomes. "This awakening" is not "random awakening" - awakenings in the experiment do not happen at random, they have order: Tails&Monday is always followed by Tails&Tuesday. Neither you can say that "this awakening" is "any awakening" because for first and second awakenings probability that the coin is Heads is different.
To reason correctly about the problem you need to talk about events that happen in this experiment, not in illdefined "this awakening". Aplying the same principle here we get:
P(Lab) = 1; P(Heads|Lab) = P(Tails|Lab) =1/2 - regardless of the outcome of the toss, in every experiment you will be awakened in the lab, so finding yourself in the lab in this experiment doesn't tell you anything about the state of the coin.
P(Darkness) = 1 P(Heads|Darkness) = P(Tails|Darkness) =1/2 - regardless of the outcome of the toss in every experiment you find yourself with closed eyes thinking about anthropics and it tells you nothing about the state of the coin.
P(Home) = 1/2; P(Heads|Home) = 1 - you find yourself home after a memory loss in every experiment only when the coin is Heads so you update in favor of it.
I [am a thirder, but] have flirted with halfing and can clearly see the attraction. As such, I don’t quite see how the conclusions you reach in your version of the experiment transfer over to the original version?
If I understand your version, you are giving the subject an equal number of observations (wake-ups) regardless of the outcome of the coin flip? That seems to materially change the experiment (whether or not the subject’s eyes are open). To me at least, the thing that makes the problem strain my intuition is exactly the mismatch between the uneven number of samplings/observations and the 50/50 coin flip. Even it out, and there’s nothing more to solve. I guess that’s maybe the point… To create a scenario that’s easier to parse, but I think something more is lost along the way.
[EDIT: I deleted the rest of this comment, as I had my mind challenged / changed enough by Ape in the coat, below, that I didn’t want to leave it up.]
> "However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails."
Before opening your eyes, you should presumably have 25% credence in each of the four outcomes (Heads-Day1-lab, Heads-Day2-bed, Tails-Day1-lab, Tails-Day2-lab). If you're in the lab, that rules out the second of the four listed possibilities. But why assume that its probability must be redistributed equally amongst the remaining three? Wouldn't it seem more natural for a Halfer to assign (conditional on being in the lab): 50% Heads-day1, 25%-Tails-day1, 25%-Tails-day2?
Perhaps the issue is that Halfers seem to be precisely those who are inclined to reject the sort of anthropic reasoning you rely on here. So it shouldn't be surprising that inconsistencies result from combining Halfing with anthropic reasoning of that kind.
> However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails.
This is wrong.
As I've explained here
https://www.lesswrong.com/posts/gwfgFwrrYnDpcF4JP/the-solution-to-sleeping-beauty
Beauty can't lawfully reason about the problem, while treating awakenings as individual outcomes. "This awakening" is not "random awakening" - awakenings in the experiment do not happen at random, they have order: Tails&Monday is always followed by Tails&Tuesday. Neither you can say that "this awakening" is "any awakening" because for first and second awakenings probability that the coin is Heads is different.
To reason correctly about the problem you need to talk about events that happen in this experiment, not in illdefined "this awakening". Aplying the same principle here we get:
P(Lab) = 1; P(Heads|Lab) = P(Tails|Lab) =1/2 - regardless of the outcome of the toss, in every experiment you will be awakened in the lab, so finding yourself in the lab in this experiment doesn't tell you anything about the state of the coin.
P(Darkness) = 1 P(Heads|Darkness) = P(Tails|Darkness) =1/2 - regardless of the outcome of the toss in every experiment you find yourself with closed eyes thinking about anthropics and it tells you nothing about the state of the coin.
P(Home) = 1/2; P(Heads|Home) = 1 - you find yourself home after a memory loss in every experiment only when the coin is Heads so you update in favor of it.
And so everything adds up to normality.
I [am a thirder, but] have flirted with halfing and can clearly see the attraction. As such, I don’t quite see how the conclusions you reach in your version of the experiment transfer over to the original version?
If I understand your version, you are giving the subject an equal number of observations (wake-ups) regardless of the outcome of the coin flip? That seems to materially change the experiment (whether or not the subject’s eyes are open). To me at least, the thing that makes the problem strain my intuition is exactly the mismatch between the uneven number of samplings/observations and the 50/50 coin flip. Even it out, and there’s nothing more to solve. I guess that’s maybe the point… To create a scenario that’s easier to parse, but I think something more is lost along the way.
[EDIT: I deleted the rest of this comment, as I had my mind challenged / changed enough by Ape in the coat, below, that I didn’t want to leave it up.]
> "However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails."
Before opening your eyes, you should presumably have 25% credence in each of the four outcomes (Heads-Day1-lab, Heads-Day2-bed, Tails-Day1-lab, Tails-Day2-lab). If you're in the lab, that rules out the second of the four listed possibilities. But why assume that its probability must be redistributed equally amongst the remaining three? Wouldn't it seem more natural for a Halfer to assign (conditional on being in the lab): 50% Heads-day1, 25%-Tails-day1, 25%-Tails-day2?
Perhaps the issue is that Halfers seem to be precisely those who are inclined to reject the sort of anthropic reasoning you rely on here. So it shouldn't be surprising that inconsistencies result from combining Halfing with anthropic reasoning of that kind.