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The Sleeping Beauty Problem

dylancatlow
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2/3/2015 12:08:10 PM
Posted: 1 year ago
The 1/2 position is pretty straightforward and easy to understand: the chance that the coin was heads is 50/50, since that's simply how coins work. The 1/3 position asserts that if the coin landed on tails, there would be a greater chance that a given experience belongs to the outcome with more "experiences" overall.
Garbanza
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2/3/2015 5:51:29 PM
Posted: 1 year ago
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.
dylancatlow
Posts: 12,242
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2/3/2015 5:59:24 PM
Posted: 1 year ago
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.
Garbanza
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2/3/2015 6:07:36 PM
Posted: 1 year ago
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.

Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.

I know that's not really answering you question, but it's all i've got right now.
bossyburrito
Posts: 14,075
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2/3/2015 9:54:30 PM
Posted: 1 year ago
At 2/3/2015 6:07:36 PM, Garbanza wrote:
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.
I don't think that's a comparable analogy, given the fact that in the thought experiment it is the same person that's woken up. No matter what was flipped, there is no chance that she wouldn't be woken up. As such, I don't see why the addition of a second wakeup changes anything. There is no way to distinguish between a heads or tails wakeup - if it's the millionth wakeup, it'll be indistinguishable from the first. There is no reason to believe that it is the millionth wakeup.
Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.
It would be a more valid comparison if heads gave you one piece of candy and tails gave you 1,000,000, but you could only remember getting the last piece of that million. Either way, you're getting candy, so there's no reason to believe that you got it via heads or tails.
I know that's not really answering you question, but it's all i've got right now.
#UnbanTheMadman

"Some will sell their dreams for small desires
Or lose the race to rats
Get caught in ticking traps
And start to dream of somewhere
To relax their restless flight
Somewhere out of a memory of lighted streets on quiet nights..."

~ Rush
Garbanza
Posts: 1,997
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2/4/2015 1:46:52 AM
Posted: 1 year ago
At 2/3/2015 9:54:30 PM, bossyburrito wrote:
At 2/3/2015 6:07:36 PM, Garbanza wrote:
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.
I don't think that's a comparable analogy, given the fact that in the thought experiment it is the same person that's woken up. No matter what was flipped, there is no chance that she wouldn't be woken up. As such, I don't see why the addition of a second wakeup changes anything. There is no way to distinguish between a heads or tails wakeup - if it's the millionth wakeup, it'll be indistinguishable from the first. There is no reason to believe that it is the millionth wakeup.
Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.
It would be a more valid comparison if heads gave you one piece of candy and tails gave you 1,000,000, but you could only remember getting the last piece of that million. Either way, you're getting candy, so there's no reason to believe that you got it via heads or tails.
I know that's not really answering you question, but it's all i've got right now.

Suppose it was like THIS. If it was tails, they wake her up on Monday. If it's heads, they throw a die, and if it's a six, they wake her up, and otherwise they don't.

Summary: 100% chance of being woken up if it's tails. 1/6 chance of being woken up if it's heads. Before it happens, there's 6/12 chance that it'll be tails and she'll be woken up, 1/12 that it'll be heads and she'll be woken up, and 5/12 that it'll be heads and she'll sleep through.

Monday morning comes along (there's a calendar on the wall) and she's woken up. Those 5/12 sleep through chances are eliminated. Was the first coin toss heads or tails? Again, I'd go for tails because there's 6/7 chances that it's tails and only 1/7 chances that it's heads.

It's the same problem from HER perspective as the other one where she doesn't know what day it is, except that she doesn't have the certainty of elimination. But there's a 2/3 chance that it's Monday, and only 1/3 chance that it's Tuesday. If it's Monday, there's a 50/50 chance of being heads, and if it's Tuesday, there's a 0% chance of heads. Overall, that simplifies to a 1/3 chance of heads.
dylancatlow
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2/4/2015 1:48:15 PM
Posted: 1 year ago
At 2/3/2015 9:54:30 PM, bossyburrito wrote:

Another argument is based on long-run average outcomes. Suppose this experiment were repeated 1,000 times. It is expected that there would be 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in one-third of the cases would heads precede her awakening. This long-run expectation should give the same expectations for the one trial, so P(Heads) = 1/3.
bossyburrito
Posts: 14,075
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2/4/2015 7:38:51 PM
Posted: 1 year ago
At 2/4/2015 1:48:15 PM, dylancatlow wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:

Another argument is based on long-run average outcomes. Suppose this experiment were repeated 1,000 times. It is expected that there would be 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in one-third of the cases would heads precede her awakening. This long-run expectation should give the same expectations for the one trial, so P(Heads) = 1/3.

Her being woken up 500 times on Monday when the coin flips tails is irrelevant when Beauty will only remember one instance of being woken up. The fact of the matter is that, if the coin lands heads, she'll remember being woken up once - if it flips tails, she'll remember being woken up once. It doesn't matter if she woke up on both Monday and Tuesday - from her perspective, no new information is gained from the forgotten waking.

Let me frame the question in a different way: if heads are flipped, she gets woken up 500 times. If tails are flipped, the experimenter or whoever shoots 500 birds and then, from the next day onward, she gets woken up 500 times. So what if 500 birds are shot? How is that relevant to the coin flip? The 500 birds will have exactly the same significance on her knowledge of the situation (that she was woken up at least once) as her being woken up 500 times on Monday and 500 times on Tuesday.

It's still a fifty-fifty split between 500 and 1000 wakings - nothing changes just because the number of wakings increases. 1000 wakings has the same probability of occurring as 1000, so there is no increased probability of the 1000 wakings having occurred.
#UnbanTheMadman

"Some will sell their dreams for small desires
Or lose the race to rats
Get caught in ticking traps
And start to dream of somewhere
To relax their restless flight
Somewhere out of a memory of lighted streets on quiet nights..."

~ Rush
bossyburrito
Posts: 14,075
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2/4/2015 7:41:07 PM
Posted: 1 year ago
At 2/4/2015 1:46:52 AM, Garbanza wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:
At 2/3/2015 6:07:36 PM, Garbanza wrote:
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.
I don't think that's a comparable analogy, given the fact that in the thought experiment it is the same person that's woken up. No matter what was flipped, there is no chance that she wouldn't be woken up. As such, I don't see why the addition of a second wakeup changes anything. There is no way to distinguish between a heads or tails wakeup - if it's the millionth wakeup, it'll be indistinguishable from the first. There is no reason to believe that it is the millionth wakeup.
Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.
It would be a more valid comparison if heads gave you one piece of candy and tails gave you 1,000,000, but you could only remember getting the last piece of that million. Either way, you're getting candy, so there's no reason to believe that you got it via heads or tails.
I know that's not really answering you question, but it's all i've got right now.

Suppose it was like THIS. If it was tails, they wake her up on Monday. If it's heads, they throw a die, and if it's a six, they wake her up, and otherwise they don't.

Summary: 100% chance of being woken up if it's tails. 1/6 chance of being woken up if it's heads. Before it happens, there's 6/12 chance that it'll be tails and she'll be woken up, 1/12 that it'll be heads and she'll be woken up, and 5/12 that it'll be heads and she'll sleep through.

Monday morning comes along (there's a calendar on the wall) and she's woken up. Those 5/12 sleep through chances are eliminated. Was the first coin toss heads or tails? Again, I'd go for tails because there's 6/7 chances that it's tails and only 1/7 chances that it's heads.

It's the same problem from HER perspective as the other one where she doesn't know what day it is, except that she doesn't have the certainty of elimination. But there's a 2/3 chance that it's Monday, and only 1/3 chance that it's Tuesday. If it's Monday, there's a 50/50 chance of being heads, and if it's Tuesday, there's a 0% chance of heads. Overall, that simplifies to a 1/3 chance of heads.

There is no such dice being cast, though. She knows, with 100% certainty, that she was woken up at least once. She knows nothing else that's relevant - if it's 50/50 between 10 and 20 wakings, the 20 wakings doesn't suddenly become more likely than the ten just because she woke up once.
#UnbanTheMadman

"Some will sell their dreams for small desires
Or lose the race to rats
Get caught in ticking traps
And start to dream of somewhere
To relax their restless flight
Somewhere out of a memory of lighted streets on quiet nights..."

~ Rush
Garbanza
Posts: 1,997
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2/4/2015 7:52:08 PM
Posted: 1 year ago
At 2/4/2015 7:41:07 PM, bossyburrito wrote:
At 2/4/2015 1:46:52 AM, Garbanza wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:
At 2/3/2015 6:07:36 PM, Garbanza wrote:
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.
I don't think that's a comparable analogy, given the fact that in the thought experiment it is the same person that's woken up. No matter what was flipped, there is no chance that she wouldn't be woken up. As such, I don't see why the addition of a second wakeup changes anything. There is no way to distinguish between a heads or tails wakeup - if it's the millionth wakeup, it'll be indistinguishable from the first. There is no reason to believe that it is the millionth wakeup.
Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.
It would be a more valid comparison if heads gave you one piece of candy and tails gave you 1,000,000, but you could only remember getting the last piece of that million. Either way, you're getting candy, so there's no reason to believe that you got it via heads or tails.
I know that's not really answering you question, but it's all i've got right now.

Suppose it was like THIS. If it was tails, they wake her up on Monday. If it's heads, they throw a die, and if it's a six, they wake her up, and otherwise they don't.

Summary: 100% chance of being woken up if it's tails. 1/6 chance of being woken up if it's heads. Before it happens, there's 6/12 chance that it'll be tails and she'll be woken up, 1/12 that it'll be heads and she'll be woken up, and 5/12 that it'll be heads and she'll sleep through.

Monday morning comes along (there's a calendar on the wall) and she's woken up. Those 5/12 sleep through chances are eliminated. Was the first coin toss heads or tails? Again, I'd go for tails because there's 6/7 chances that it's tails and only 1/7 chances that it's heads.

It's the same problem from HER perspective as the other one where she doesn't know what day it is, except that she doesn't have the certainty of elimination. But there's a 2/3 chance that it's Monday, and only 1/3 chance that it's Tuesday. If it's Monday, there's a 50/50 chance of being heads, and if it's Tuesday, there's a 0% chance of heads. Overall, that simplifies to a 1/3 chance of heads.

There is no such dice being cast, though. She knows, with 100% certainty, that she was woken up at least once. She knows nothing else that's relevant - if it's 50/50 between 10 and 20 wakings, the 20 wakings doesn't suddenly become more likely than the ten just because she woke up once.

Yes it does, If she's just woken up, it's twice as likely to be on tails as it is on heads. It's like dylancatlow said. Imagine they ran the experiment 20 times, and it fell heads 10 times and tails 10 times. If she guessed heads 50% of the time, she'd be right 15 times out of 30 wakenings. If she guessed tails ALL the time, she'd be right 20 times out of 30. So actually, guessing tails all the time is the best strategy and that couldn't be the case if there really was a 50/50 chance of heads or tails.
bossyburrito
Posts: 14,075
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2/4/2015 7:58:28 PM
Posted: 1 year ago
At 2/4/2015 7:52:08 PM, Garbanza wrote:
At 2/4/2015 7:41:07 PM, bossyburrito wrote:
At 2/4/2015 1:46:52 AM, Garbanza wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:
At 2/3/2015 6:07:36 PM, Garbanza wrote:
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.
I don't think that's a comparable analogy, given the fact that in the thought experiment it is the same person that's woken up. No matter what was flipped, there is no chance that she wouldn't be woken up. As such, I don't see why the addition of a second wakeup changes anything. There is no way to distinguish between a heads or tails wakeup - if it's the millionth wakeup, it'll be indistinguishable from the first. There is no reason to believe that it is the millionth wakeup.
Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.
It would be a more valid comparison if heads gave you one piece of candy and tails gave you 1,000,000, but you could only remember getting the last piece of that million. Either way, you're getting candy, so there's no reason to believe that you got it via heads or tails.
I know that's not really answering you question, but it's all i've got right now.

Suppose it was like THIS. If it was tails, they wake her up on Monday. If it's heads, they throw a die, and if it's a six, they wake her up, and otherwise they don't.

Summary: 100% chance of being woken up if it's tails. 1/6 chance of being woken up if it's heads. Before it happens, there's 6/12 chance that it'll be tails and she'll be woken up, 1/12 that it'll be heads and she'll be woken up, and 5/12 that it'll be heads and she'll sleep through.

Monday morning comes along (there's a calendar on the wall) and she's woken up. Those 5/12 sleep through chances are eliminated. Was the first coin toss heads or tails? Again, I'd go for tails because there's 6/7 chances that it's tails and only 1/7 chances that it's heads.

It's the same problem from HER perspective as the other one where she doesn't know what day it is, except that she doesn't have the certainty of elimination. But there's a 2/3 chance that it's Monday, and only 1/3 chance that it's Tuesday. If it's Monday, there's a 50/50 chance of being heads, and if it's Tuesday, there's a 0% chance of heads. Overall, that simplifies to a 1/3 chance of heads.

There is no such dice being cast, though. She knows, with 100% certainty, that she was woken up at least once. She knows nothing else that's relevant - if it's 50/50 between 10 and 20 wakings, the 20 wakings doesn't suddenly become more likely than the ten just because she woke up once.

Yes it does, If she's just woken up, it's twice as likely to be on tails as it is on heads.
No, it's not. The whole thing is determined the moment the coin is flipped - that initial 50/50 flip determines whether there's even the possibility of a second waking or not. It doesn't matter if it's possible if it's constrained at the outset by the need of the coinflip.

It's like dylancatlow said. Imagine they ran the experiment 20 times, and it fell heads 10 times and tails 10 times. If she guessed heads 50% of the time, she'd be right 15 times out of 30 wakenings. If she guessed tails ALL the time, she'd be right 20 times out of 30. So actually, guessing tails all the time is the best strategy and that couldn't be the case if there really was a 50/50 chance of heads or tails.
That literally is just assuming the conclusion that it's a higher chance of tails, though. It doesn't validate anything because it itself isn't justified.
#UnbanTheMadman

"Some will sell their dreams for small desires
Or lose the race to rats
Get caught in ticking traps
And start to dream of somewhere
To relax their restless flight
Somewhere out of a memory of lighted streets on quiet nights..."

~ Rush
Garbanza
Posts: 1,997
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2/4/2015 8:02:51 PM
Posted: 1 year ago
At 2/4/2015 7:58:28 PM, bossyburrito wrote:
At 2/4/2015 7:52:08 PM, Garbanza wrote:
At 2/4/2015 7:41:07 PM, bossyburrito wrote:
At 2/4/2015 1:46:52 AM, Garbanza wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:
At 2/3/2015 6:07:36 PM, Garbanza wrote:
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.
I don't think that's a comparable analogy, given the fact that in the thought experiment it is the same person that's woken up. No matter what was flipped, there is no chance that she wouldn't be woken up. As such, I don't see why the addition of a second wakeup changes anything. There is no way to distinguish between a heads or tails wakeup - if it's the millionth wakeup, it'll be indistinguishable from the first. There is no reason to believe that it is the millionth wakeup.
Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.
It would be a more valid comparison if heads gave you one piece of candy and tails gave you 1,000,000, but you could only remember getting the last piece of that million. Either way, you're getting candy, so there's no reason to believe that you got it via heads or tails.
I know that's not really answering you question, but it's all i've got right now.

Suppose it was like THIS. If it was tails, they wake her up on Monday. If it's heads, they throw a die, and if it's a six, they wake her up, and otherwise they don't.

Summary: 100% chance of being woken up if it's tails. 1/6 chance of being woken up if it's heads. Before it happens, there's 6/12 chance that it'll be tails and she'll be woken up, 1/12 that it'll be heads and she'll be woken up, and 5/12 that it'll be heads and she'll sleep through.

Monday morning comes along (there's a calendar on the wall) and she's woken up. Those 5/12 sleep through chances are eliminated. Was the first coin toss heads or tails? Again, I'd go for tails because there's 6/7 chances that it's tails and only 1/7 chances that it's heads.

It's the same problem from HER perspective as the other one where she doesn't know what day it is, except that she doesn't have the certainty of elimination. But there's a 2/3 chance that it's Monday, and only 1/3 chance that it's Tuesday. If it's Monday, there's a 50/50 chance of being heads, and if it's Tuesday, there's a 0% chance of heads. Overall, that simplifies to a 1/3 chance of heads.

There is no such dice being cast, though. She knows, with 100% certainty, that she was woken up at least once. She knows nothing else that's relevant - if it's 50/50 between 10 and 20 wakings, the 20 wakings doesn't suddenly become more likely than the ten just because she woke up once.

Yes it does, If she's just woken up, it's twice as likely to be on tails as it is on heads.
No, it's not. The whole thing is determined the moment the coin is flipped - that initial 50/50 flip determines whether there's even the possibility of a second waking or not. It doesn't matter if it's possible if it's constrained at the outset by the need of the coinflip.

Could you explain this again? I don't understand what you mean.

It's like dylancatlow said. Imagine they ran the experiment 20 times, and it fell heads 10 times and tails 10 times. If she guessed heads 50% of the time, she'd be right 15 times out of 30 wakenings. If she guessed tails ALL the time, she'd be right 20 times out of 30. So actually, guessing tails all the time is the best strategy and that couldn't be the case if there really was a 50/50 chance of heads or tails.
That literally is just assuming the conclusion that it's a higher chance of tails, though. It doesn't validate anything because it itself isn't justified.

No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.
bossyburrito
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2/4/2015 8:08:55 PM
Posted: 1 year ago
At 2/4/2015 8:02:51 PM, Garbanza wrote:
At 2/4/2015 7:58:28 PM, bossyburrito wrote:
At 2/4/2015 7:52:08 PM, Garbanza wrote:
At 2/4/2015 7:41:07 PM, bossyburrito wrote:
At 2/4/2015 1:46:52 AM, Garbanza wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:
At 2/3/2015 6:07:36 PM, Garbanza wrote:
At 2/3/2015 5:59:24 PM, dylancatlow wrote:
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

You do realize this is meant to be a paradox, right? It's not enough to explain why one side makes sense, you have to explain why the other side is wrong.

Idk. I think of each being woken up as a separate person. So, there's a game where if it's heads, one random student gets given free candy, and if it's tails, a hundred random students get given candy. You're walking around on campus one day, and someone gives you free candy, is it more likely to be heads or tails? Obviously, tails.
I don't think that's a comparable analogy, given the fact that in the thought experiment it is the same person that's woken up. No matter what was flipped, there is no chance that she wouldn't be woken up. As such, I don't see why the addition of a second wakeup changes anything. There is no way to distinguish between a heads or tails wakeup - if it's the millionth wakeup, it'll be indistinguishable from the first. There is no reason to believe that it is the millionth wakeup.
Like, the original coin toss had a 50/50 chance of being heads or tails, but the fact that you're getting candy is evidence for tails, so you'd guess tails.
It would be a more valid comparison if heads gave you one piece of candy and tails gave you 1,000,000, but you could only remember getting the last piece of that million. Either way, you're getting candy, so there's no reason to believe that you got it via heads or tails.
I know that's not really answering you question, but it's all i've got right now.

Suppose it was like THIS. If it was tails, they wake her up on Monday. If it's heads, they throw a die, and if it's a six, they wake her up, and otherwise they don't.

Summary: 100% chance of being woken up if it's tails. 1/6 chance of being woken up if it's heads. Before it happens, there's 6/12 chance that it'll be tails and she'll be woken up, 1/12 that it'll be heads and she'll be woken up, and 5/12 that it'll be heads and she'll sleep through.

Monday morning comes along (there's a calendar on the wall) and she's woken up. Those 5/12 sleep through chances are eliminated. Was the first coin toss heads or tails? Again, I'd go for tails because there's 6/7 chances that it's tails and only 1/7 chances that it's heads.

It's the same problem from HER perspective as the other one where she doesn't know what day it is, except that she doesn't have the certainty of elimination. But there's a 2/3 chance that it's Monday, and only 1/3 chance that it's Tuesday. If it's Monday, there's a 50/50 chance of being heads, and if it's Tuesday, there's a 0% chance of heads. Overall, that simplifies to a 1/3 chance of heads.

There is no such dice being cast, though. She knows, with 100% certainty, that she was woken up at least once. She knows nothing else that's relevant - if it's 50/50 between 10 and 20 wakings, the 20 wakings doesn't suddenly become more likely than the ten just because she woke up once.

Yes it does, If she's just woken up, it's twice as likely to be on tails as it is on heads.
No, it's not. The whole thing is determined the moment the coin is flipped - that initial 50/50 flip determines whether there's even the possibility of a second waking or not. It doesn't matter if it's possible if it's constrained at the outset by the need of the coinflip.

Could you explain this again? I don't understand what you mean.
At the outset, there are two possibilities: heads is flipped and she wakes up, say, ten times, or tails is flipped and she wakes up 20 times. This does NOT mean that she wakes up 20 times regardless - she only does so in the case of tails being flipped. As such, to use the possibility of the 20 wakeups, you have to accept, at the start, that the 20 wakeups itself has a 50% probability of occurring at all. No matter how many times she wakes up, it all depends on the first coin flip. : : :It's like dylancatlow said. Imagine they ran the experiment 20 times, and it fell heads 10 times and tails 10 times. If she guessed heads 50% of the time, she'd be right 15 times out of 30 wakenings. If she guessed tails ALL the time, she'd be right 20 times out of 30. So actually, guessing tails all the time is the best strategy and that couldn't be the case if there really was a 50/50 chance of heads or tails.
That literally is just assuming the conclusion that it's a higher chance of tails, though. It doesn't validate anything because it itself isn't justified.

No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.
She only wakes up 10 extra times on tails when tails is flipped. She doesn't do that in all scenarios - those scenarios with the 10 extra wakeups can only exist when tails has already been flipped, and that flip depends on a 50/50 chance.
#UnbanTheMadman

"Some will sell their dreams for small desires
Or lose the race to rats
Get caught in ticking traps
And start to dream of somewhere
To relax their restless flight
Somewhere out of a memory of lighted streets on quiet nights..."

~ Rush
Garbanza
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2/4/2015 8:36:04 PM
Posted: 1 year ago
At 2/4/2015 8:08:55 PM, bossyburrito wrote:
No, it's not. The whole thing is determined the moment the coin is flipped - that initial 50/50 flip determines whether there's even the possibility of a second waking or not. It doesn't matter if it's possible if it's constrained at the outset by the need of the coinflip.

Could you explain this again? I don't understand what you mean.
At the outset, there are two possibilities: heads is flipped and she wakes up, say, ten times, or tails is flipped and she wakes up 20 times. This does NOT mean that she wakes up 20 times regardless - she only does so in the case of tails being flipped. As such, to use the possibility of the 20 wakeups, you have to accept, at the start, that the 20 wakeups itself has a 50% probability of occurring at all. No matter how many times she wakes up, it all depends on the first coin flip.

Right. There's a 50/50 chance of heads or tails, but for each wake up situation, there's a 2/3 chance of tails.

It's like dylancatlow said. Imagine they ran the experiment 20 times, and it fell heads 10 times and tails 10 times. If she guessed heads 50% of the time, she'd be right 15 times out of 30 wakenings. If she guessed tails ALL the time, she'd be right 20 times out of 30. So actually, guessing tails all the time is the best strategy and that couldn't be the case if there really was a 50/50 chance of heads or tails.
That literally is just assuming the conclusion that it's a higher chance of tails, though. It doesn't validate anything because it itself isn't justified.

No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.
She only wakes up 10 extra times on tails when tails is flipped. She doesn't do that in all scenarios - those scenarios with the 10 extra wakeups can only exist when tails has already been flipped, and that flip depends on a 50/50 chance.

Right. (?)
bossyburrito
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2/4/2015 8:40:18 PM
Posted: 1 year ago
At 2/4/2015 8:36:04 PM, Garbanza wrote:
At 2/4/2015 8:08:55 PM, bossyburrito wrote:
No, it's not. The whole thing is determined the moment the coin is flipped - that initial 50/50 flip determines whether there's even the possibility of a second waking or not. It doesn't matter if it's possible if it's constrained at the outset by the need of the coinflip.

Could you explain this again? I don't understand what you mean.
At the outset, there are two possibilities: heads is flipped and she wakes up, say, ten times, or tails is flipped and she wakes up 20 times. This does NOT mean that she wakes up 20 times regardless - she only does so in the case of tails being flipped. As such, to use the possibility of the 20 wakeups, you have to accept, at the start, that the 20 wakeups itself has a 50% probability of occurring at all. No matter how many times she wakes up, it all depends on the first coin flip.

Right. There's a 50/50 chance of heads or tails, but for each wake up situation, there's a 2/3 chance of tails.
There's a 2/3 chance of tails if there's a 2/3 chance of tails, but there's not. There's a 50/50 chance of tails.
It's like dylancatlow said. Imagine they ran the experiment 20 times, and it fell heads 10 times and tails 10 times. If she guessed heads 50% of the time, she'd be right 15 times out of 30 wakenings. If she guessed tails ALL the time, she'd be right 20 times out of 30. So actually, guessing tails all the time is the best strategy and that couldn't be the case if there really was a 50/50 chance of heads or tails.
That literally is just assuming the conclusion that it's a higher chance of tails, though. It doesn't validate anything because it itself isn't justified.

No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.
She only wakes up 10 extra times on tails when tails is flipped. She doesn't do that in all scenarios - those scenarios with the 10 extra wakeups can only exist when tails has already been flipped, and that flip depends on a 50/50 chance.

Right. (?)

Are you agreeing with me?
#UnbanTheMadman

"Some will sell their dreams for small desires
Or lose the race to rats
Get caught in ticking traps
And start to dream of somewhere
To relax their restless flight
Somewhere out of a memory of lighted streets on quiet nights..."

~ Rush
Garbanza
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2/4/2015 10:06:14 PM
Posted: 1 year ago
At 2/4/2015 8:40:18 PM, bossyburrito wrote:
At 2/4/2015 8:36:04 PM, Garbanza wrote:
At 2/4/2015 8:08:55 PM, bossyburrito wrote:
No, it's not. The whole thing is determined the moment the coin is flipped - that initial 50/50 flip determines whether there's even the possibility of a second waking or not. It doesn't matter if it's possible if it's constrained at the outset by the need of the coinflip.

Could you explain this again? I don't understand what you mean.
At the outset, there are two possibilities: heads is flipped and she wakes up, say, ten times, or tails is flipped and she wakes up 20 times. This does NOT mean that she wakes up 20 times regardless - she only does so in the case of tails being flipped. As such, to use the possibility of the 20 wakeups, you have to accept, at the start, that the 20 wakeups itself has a 50% probability of occurring at all. No matter how many times she wakes up, it all depends on the first coin flip.

Right. There's a 50/50 chance of heads or tails, but for each wake up situation, there's a 2/3 chance of tails.
There's a 2/3 chance of tails if there's a 2/3 chance of tails, but there's not. There's a 50/50 chance of tails.
It's like dylancatlow said. Imagine they ran the experiment 20 times, and it fell heads 10 times and tails 10 times. If she guessed heads 50% of the time, she'd be right 15 times out of 30 wakenings. If she guessed tails ALL the time, she'd be right 20 times out of 30. So actually, guessing tails all the time is the best strategy and that couldn't be the case if there really was a 50/50 chance of heads or tails.
That literally is just assuming the conclusion that it's a higher chance of tails, though. It doesn't validate anything because it itself isn't justified.

No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.
She only wakes up 10 extra times on tails when tails is flipped. She doesn't do that in all scenarios - those scenarios with the 10 extra wakeups can only exist when tails has already been flipped, and that flip depends on a 50/50 chance.

Right. (?)

Are you agreeing with me?

I'm agreeing with what you say, but I don't agree with your conclusions, but for reasons I've already stated.
dylancatlow
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2/5/2015 2:42:05 PM
Posted: 1 year ago
At 2/4/2015 7:38:51 PM, bossyburrito wrote:
At 2/4/2015 1:48:15 PM, dylancatlow wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:

Another argument is based on long-run average outcomes. Suppose this experiment were repeated 1,000 times. It is expected that there would be 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in one-third of the cases would heads precede her awakening. This long-run expectation should give the same expectations for the one trial, so P(Heads) = 1/3.

Her being woken up 500 times on Monday when the coin flips tails is irrelevant when Beauty will only remember one instance of being woken up. The fact of the matter is that, if the coin lands heads, she'll remember being woken up once - if it flips tails, she'll remember being woken up once. It doesn't matter if she woke up on both Monday and Tuesday - from her perspective, no new information is gained from the forgotten waking.


This kind of misses the point. If you repeat the experiment 1000 times, she will be woken up, on average, 1500 times, and in 1000 of those cases, the coin flip would have been tails. So for a given experience, in that moment, there's a 2/3 chance it belongs to a situation in which the coin flip was tails. Memory is irrelevant.
dylancatlow
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2/5/2015 4:04:05 PM
Posted: 1 year ago
I think the resolution would look similar to Langan's solution for the Paradox of Kraitchik: http://www.megafoundation.org...

Basically, the reasoning needs to be made symmetrical.
Garbanza
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2/5/2015 10:04:53 PM
Posted: 1 year ago
At 2/5/2015 4:04:05 PM, dylancatlow wrote:
I think the resolution would look similar to Langan's solution for the Paradox of Kraitchik: http://www.megafoundation.org...

Basically, the reasoning needs to be made symmetrical.

Haha that's brilliant. I never saw that two envelope problem before. So if you consider your perspective, the other person averages 1.25x your value. But if you consider their perspective, they have 0.8x your value. What a mind fvck. Love it. :)

I can't figure out what it means though for the sleeping beauty problem.
dylancatlow
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2/6/2015 3:27:56 PM
Posted: 1 year ago
For those who didn't already notice, the 1/3 figure is actually predicated on the 1/2 figure, so it's obviously the answer in need of correction.
dylancatlow
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2/6/2015 4:02:06 PM
Posted: 1 year ago
I think what it boils down to is that, although the chances of you experiencing a tails outcome it greater than a heads outcome, this doesn't require that a tails outcome actually occur more often, since there are two experiences associated with a tails outcome. The fallacy follows from the assumption that there is only one subjective frame to consider - the assumption that the extra experiences associated with a tails outcome must all be accounted for "right now" - when in fact, there's a 50 percent chance that there are two subjective frames, so the extra experiences could be accounted for by another "experiencer."
sdavio
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2/6/2015 10:50:46 PM
Posted: 1 year ago
At 2/3/2015 5:51:29 PM, Garbanza wrote:
At 2/3/2015 11:47:48 AM, dylancatlow wrote:
Thoughts? http://en.wikipedia.org...

I think it has to be a third because of the extreme condition. Suppose she got woken on a Monday for heads, and then slept for a hundred days, or got woken 101 days in a row for tails. Then, if you're being woken up, it's MUCH more likely that it's a tails than heads, right? So you'd have to guess tails.

But in either condition she is under the impression of having been woken up for the first time, so there's no differentiation in that.

Also, in your "woken up once vs 101 times" situation, the chance is 1/2 once vs 1/2 101 times. It is backwards reasoning to say that it's more likely since 101 is a higher number, because the number of times woken up does not influence the coin flip. Rather, the coin flip determines the number of awakenings.
"Logic is the money of the mind." - Karl Marx
sdavio
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2/6/2015 10:54:37 PM
Posted: 1 year ago
The outcome of the coin flip of obviously 1/2 each probability. Then all that's different is the number of times she's asked about the outcome. But because that number of times is unknown to the sleeping beauty, it is an arbitrary, irrelevant variable. The possibility that she had been awoken before does not affect the probability of the coin flip.
"Logic is the money of the mind." - Karl Marx
sdavio
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2/6/2015 11:03:36 PM
Posted: 1 year ago
At 2/4/2015 1:48:15 PM, dylancatlow wrote:
At 2/3/2015 9:54:30 PM, bossyburrito wrote:

Another argument is based on long-run average outcomes. Suppose this experiment were repeated 1,000 times. It is expected that there would be 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in one-third of the cases would heads precede her awakening. This long-run expectation should give the same expectations for the one trial, so P(Heads) = 1/3.

"only in one-third of the cases would heads precede her awakening."

But the number of 'cases' (of being woken up) is arbitrary. It did not affect the outcome of the coin-flip. She doesn't know how many times she's been woken up, so she can't include it in such a mathematical formula. The chance is still 1/2 that she's been woken up the first time or the second.
"Logic is the money of the mind." - Karl Marx
sdavio
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2/6/2015 11:12:41 PM
Posted: 1 year ago
At 2/4/2015 8:02:51 PM, Garbanza wrote:
No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.

As it said in the "phenomenalist" part, if she is rewarded for each correct guess, then it would be beneficial to guess tails every time. But this is because it has a better outcome.

If I said, I'm going to flip a coin, and if it's heads and you guess correctly you get $50, and if it's tails and you guess correctly you get $100, you would guess tails no matter what, because there's a better prize for the outcome. But the outcome does not affect the probability.
"Logic is the money of the mind." - Karl Marx
Garbanza
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2/7/2015 2:51:56 AM
Posted: 1 year ago
At 2/6/2015 10:54:37 PM, sdavio wrote:
The outcome of the coin flip of obviously 1/2 each probability. Then all that's different is the number of times she's asked about the outcome. But because that number of times is unknown to the sleeping beauty, it is an arbitrary, irrelevant variable. The possibility that she had been awoken before does not affect the probability of the coin flip.

No, you're conflating two separate probabilities.

Before the coinflip, there's a 50/50 chance that it'll be heads or tails. We don't know what will happen.

But AFTER the coinflip, there's no longer a 50/50 chance of heads or tails. There's a 100% chance of one, and 0% chance of the other, because it's already happened. We (sleeping beauty) just don't know which it was. The only question is, do we have evidence that gives us information one way or the other?

It's the same difference with medical probabilities. If you're a 20 year old without cancer, there's a 35% chance (just making this up) that you'll die of cancer before you are 50. This would be based on population cancer rates, and it's a true probability because you have no cancer now and there's no way of telling the future. However, if you get a lump removed, and the doctor says there's 35% chance it's cancer, that's not true. It's either cancer or it isn't. It's ALREADY cancer or not. 100% or 0%. And probably, if the doctor was more knowledgeable, or if we knew more about cancer, there would be a way of knowing if it was cancer or not before the biopsy (or whatever the cancer test is). This second probability is based on the doctor's KNOWLEDGE, not on the actual probability of getting cancer. It's a different assessment entirely.

It's the same with the coinflip. It's already happened. If sleeping beauty wakes up on Monday, it could be heads or tails. There's no information contained in her waking up on Monday, so she could guess either way. However, if she wakes up on Tuesday, then it MUST be tails.

If she doesn't know which day it is when she wakes up, then there's a chance that it's Tuesday. That means that the probability of it being tails is greater than 50/50.
Garbanza
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2/7/2015 2:59:45 AM
Posted: 1 year ago
At 2/6/2015 11:12:41 PM, sdavio wrote:
At 2/4/2015 8:02:51 PM, Garbanza wrote:
No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.

As it said in the "phenomenalist" part, if she is rewarded for each correct guess, then it would be beneficial to guess tails every time. But this is because it has a better outcome.

If I said, I'm going to flip a coin, and if it's heads and you guess correctly you get $50, and if it's tails and you guess correctly you get $100, you would guess tails no matter what, because there's a better prize for the outcome. But the outcome does not affect the probability.

But she has to be right at each awakening, not at each coinflip, so I'm not sure how this part applies. Like, if she got a score per coinflip, then her results could be ALL FALSE, HALF RIGHT (for tails where she gets one day right, one day wrong), or ALL RIGHT. If she got a point for all right, a negative point for all wrong, and nothing for half right, then yeah she could go ahead and guess 50/50 whichever she felt like.

But it's not scored like that. She only has one awakening, and she has to guess it and that's all, so there's no better outcome prize. If it's Tuesday, she would be asleep if it had been heads, but she's awake, so it must be tails. If it's Monday, it could be either.
Garbanza
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2/7/2015 3:03:35 AM
Posted: 1 year ago
At 2/6/2015 4:02:06 PM, dylancatlow wrote:
I think what it boils down to is that, although the chances of you experiencing a tails outcome it greater than a heads outcome, this doesn't require that a tails outcome actually occur more often, since there are two experiences associated with a tails outcome.

Right. But SB is assessing the probability of her experience type, not the probability of the coinflip.

The fallacy follows from the assumption that there is only one subjective frame to consider - the assumption that the extra experiences associated with a tails outcome must all be accounted for "right now" - when in fact, there's a 50 percent chance that there are two subjective frames, so the extra experiences could be accounted for by another "experiencer."

What other experiencer? I don't get it.
sdavio
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2/7/2015 11:03:53 AM
Posted: 1 year ago
At 2/7/2015 2:59:45 AM, Garbanza wrote:
At 2/6/2015 11:12:41 PM, sdavio wrote:
At 2/4/2015 8:02:51 PM, Garbanza wrote:
No, but I'm saying if the coinflip comes down heads and tails exactly 50/50, even so, out of 20 trials, that's 20 wakenings on tails and 10 wakenings on heads. So if you were doing repeated trials and you wanted to be as accurate as you could be on as many wakenings as possible, you'd always pick tails. That means that for any wakening, with no extra information, the chance of it being tails is higher than the chance of it being heads.

As it said in the "phenomenalist" part, if she is rewarded for each correct guess, then it would be beneficial to guess tails every time. But this is because it has a better outcome.

If I said, I'm going to flip a coin, and if it's heads and you guess correctly you get $50, and if it's tails and you guess correctly you get $100, you would guess tails no matter what, because there's a better prize for the outcome. But the outcome does not affect the probability.

But she has to be right at each awakening, not at each coinflip, so I'm not sure how this part applies. Like, if she got a score per coinflip, then her results could be ALL FALSE, HALF RIGHT (for tails where she gets one day right, one day wrong), or ALL RIGHT. If she got a point for all right, a negative point for all wrong, and nothing for half right, then yeah she could go ahead and guess 50/50 whichever she felt like.

But it's not scored like that. She only has one awakening, and she has to guess it and that's all, so there's no better outcome prize. If it's Tuesday, she would be asleep if it had been heads, but she's awake, so it must be tails. If it's Monday, it could be either.

Looking back at the wiki page, I think part of the problem here is that it's left ambiguous as to what the 'problem' even is. What we have is a storyline, but no question is posed. So we can look at it more like a gambling game, like above, which certainly seems more interesting, however in the wiki itself it seems to say that it's a question of the sleeping beauty's actual 'belief' about the outcome of the coin toss:

"...an ideally rational epistemic agent is to be woken once or twice according to the toss of a coin, and asked her degree of belief for the coin having come up heads."

"Any time Sleeping Beauty is wakened and interviewed, she is asked, "What is your belief now for the proposition that the coin landed heads?""

So, in absence of evidence about the day, if we're just talking about her belief, the number of correct answers wouldn't come into it. We'd just be talking about the probability of the coin toss itself, in which case it seems almost trivial that it would be 50/50.

The other way which people seem to be getting around that, is by making it some kind of probability of "experiences", where she's measuring the probability of which day she's waking up, 'monday and heads', 'monday and tails', or 'tuesday and tails'. Thus, for the same reason that if she were gambling it would be prudent to bet with tails, the probability of it being a 'tails day' is 2/3.

However, in this second case, we would be answering a different question than one about her belief about the coin toss. We would be determining her belief about the day, if anything.
"Logic is the money of the mind." - Karl Marx