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Can someone explain the half position clearly? I don't understand how anyone could think half is the correct answer. Let me frame the question a different way.

Can someone explain the half position clearly? I don't understand how anyone could think half is the correct answer. Let me frame the question a different way. You are one of three volunteers in separate rooms. I flip a coin and if it's heads I ask one volunteer (at random) to guess the outcome. If I flip tails I ask two of the volunteers (again, at random) to guess the outcome. You know the rules I will follow, but you cannot tell if anyone else has been asked before you. I open the door and ask you to guess the outcome of the flip. What do you guess?

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- tantalor 11y ago
  > the question asked about the odds of the coin toss No, the question asked about the outcome of the coin toss, which is much less likely to be Heads (1 wake) than Tails (2 wakes).
- Bartweiss 11y ago
  I think the issue here is that the two of you are answering subtly different questions - which statement of the problem encourages to arrive at a 'paradox'. In particular, we need to agree that Sleeping Beauty is questioned both times on a result of tails to make progress. The coin is fair, so the chance of getting heads is obviously 1/2. That's the answer to "What are the odds that a given coin flip came up heads?", but it's not what Sleeping Beauty is being asked when she wakes up. What SB is being asked is "Given that you have just been awoken, what are the odds that you were awoken as a result of the coin landing heads?" There's an implied conditional present. The coin flip was fair, but the questioning is biased by result - if SB is awake and being asked, the odds are 2/3 that she's in a world where the coin came up tails. To demonstrate, resort to extreme cases, as you did. If SB isn't woken at all for heads, then the answer to the question "you're awake, how did the coin land?" is tails with 100% certainty. If she's woken up once for heads and 99 times for tails, then the answer to "you're awake, how did the coin land?" is tails with 99% certainty, and so on. This all hinges on the assertion that SB is questioned on every awakening. If she's only asked on the first awakening (or is asked about the coin flip as a general concept) then the answer is of course 1/2, as you assert.
  - timv 11y ago
    And herein lies the problem. You say Given that you have just been awoken, what are the odds that you were awoken as a result of the coin landing heads? But the linked post says (ambiguously) When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads? I don't think your interpretation of the post's question is accurate, but it's worded in vague enough terms that in effect we're arguing about the writer's intent rather than stats.
tantalor 11y ago
I like this rephrasing, but I think you only need n=2 subjects not n=3. To see why you should guess tails, set n=100.
CamperBob2 11y ago
The fact is, there aren't three volunteers, there's only one. Upon awakening, she has no information she didn't have when she went to sleep, which at first seems to be an unassailable argument in favor of the "halfers." To me, the problem with the "no new information" argument is that there's another popular paradox in which it also seems to apply, but fails spectacularly. In the Monty Hall paradox (https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem), you know in advance that the host is going to open a door at random and offer you a chance to change your initial guess. You're aware that the host knows what's behind the doors. You know that no matter what door you pick, one of the remaining ones has a goat behind it. And you know that the omniscient host will pick that door. So what new information do you gain when the host does what you knew he was going to do? One way to resolve the Monty Hall paradox is to rephrase it such that there are a million doors and the host opens all of the remaining ones except one. Of course it makes sense to switch doors in that scenario, since you knew that the host would avoid opening your initially-chosen door as well as the one with the car behind it. Your initial door has 1:1000000 odds of winning, while the only other closed door has odds of 1/2. So if you could find a way to transform the Sleeping Beauty problem into a Monty Hall problem, that could be a good way to settle the argument in favor of the "thirders." Or, if with some logical sleight-of-hand, you could show that SB is not isomorphic to MH, that could potentially be used to strengthen the no-new-information argument made by the "halfer" side. I'm leaning that way, because SB does not know upon awakening, and will not remember, whether the experiment will result in her being awakened twice. For all she knows, she's playing the same Monty Hall game for the second time. It's this lack of an unbroken timeline of conscious awareness that settles the question in favor of the halfers. To answer the thirders' rebuttal that SB would lose money if she were gambling on the coin toss with any odds other than 1:3, I'd say that's a completely different experiment. We aren't being asked to speculate on the set of events behind multiple awakenings, just one in isolation. The lack-of-new-information argument carries the day. Any arguments that involve repetition, clones, or gambling fail because they are responding to a different question. TL,DR: Don't gamble if the game involves erasing the players' memories. You will lose a lot of money in a hurry, even if your strategy is mathematically optimal.
- URSpider94 11y ago
te_platt 11y ago
Your reframing of the question changes the problem. Let A be the probability of Heads (1/2) and B be the probability you are asked (1/2) then P(B|A) is 1/3. So the probability the coin came up heads is P(A|B) = P(B|A) * P(A)/P(B) = 1/3. Which is what you expected. In the original problem the probability of getting woken up is 1. So we end up with the probability of the coin being heads given that I was just woken up is (1 * 1/2) /1 = 1/2.
- Retric 11y ago
  Suppose I am going to flip a coin and give you money if you guess correctly. On heads I hand you 1$ on tails I hand you 1$ and add 1$ to your bank account without telling you. Now, after flipping the coin I ask you what the heads or tails? Clearly your going to pick tails. Why? Because it's the more rational choice based on the spoken rules. Going to sleep and losing memory is really just window dressing. The simple truth is before you went to sleep you should be able to say 1/3 odds of waking up to heads.
  - timv 11y ago
    But that's still not the question. The stated question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads? Not, Should you _guess_ Heads or Tails? The probability of the coin toss _does not change_. The probability that "the reason they are waking you up is because the coin was Heads" does change, but that's not the question.
    - Retric 11y ago
      Clearly there is a 50:50 shot you get heads or tails from the coin. In your model of it's a 50:50 coin then you would have 50:25:25. The problem is waking up the second time does not reduce the odds of waking up the first time thus 50:50:50.
  - te_platt 11y ago
    I think you changed the question. Before you go to sleep you know you have a 50% chance of waking up to heads and a 50% chance of waking up to tails twice. When you wake up you haven't learned anything so those probabilities can't have changed.
logicallee 11y ago
This is an incredibly powerful rephrasing, kudos. It's perfect, because for all intents and purposes you are "a[nother] person, who cannot tell if anyone [he] has been asked before", even though that other person is "also you." Very good! The half position is clearly, simply wrong.
mokus 11y ago
I think 1/2 argument makes essential use of the fact that the "2 people" are the same person at different times. Ultimately it comes down to a different idea of the way "being right twice" is valued. In the 1/3 position, the model is that each time you answer the question you get something good if you're right. In the 1/2 perspective, the idea is that after the whole experiment is over you either were simply correct or incorrect, and being correct twice doesn't count any extra.
- timv 11y ago
  You don't get to be "right twice" The question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads? It's not a guessing competition, it's asking you to assess the probability of a past event that you have no new information about.

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tantalor 11y ago
> the question asked about the odds of the coin toss No, the question asked about the outcome of the coin toss, which is much less likely to be Heads (1 wake) than Tails (2 wakes).
Bartweiss 11y ago
I think the issue here is that the two of you are answering subtly different questions - which statement of the problem encourages to arrive at a 'paradox'. In particular, we need to agree that Sleeping Beauty is questioned both times on a result of tails to make progress. The coin is fair, so the chance of getting heads is obviously 1/2. That's the answer to "What are the odds that a given coin flip came up heads?", but it's not what Sleeping Beauty is being asked when she wakes up. What SB is being asked is "Given that you have just been awoken, what are the odds that you were awoken as a result of the coin landing heads?" There's an implied conditional present. The coin flip was fair, but the questioning is biased by result - if SB is awake and being asked, the odds are 2/3 that she's in a world where the coin came up tails. To demonstrate, resort to extreme cases, as you did. If SB isn't woken at all for heads, then the answer to the question "you're awake, how did the coin land?" is tails with 100% certainty. If she's woken up once for heads and 99 times for tails, then the answer to "you're awake, how did the coin land?" is tails with 99% certainty, and so on. This all hinges on the assertion that SB is questioned on every awakening. If she's only asked on the first awakening (or is asked about the coin flip as a general concept) then the answer is of course 1/2, as you assert.
- timv 11y ago
  And herein lies the problem. You say Given that you have just been awoken, what are the odds that you were awoken as a result of the coin landing heads? But the linked post says (ambiguously) When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads? I don't think your interpretation of the post's question is accurate, but it's worded in vague enough terms that in effect we're arguing about the writer's intent rather than stats.

tantalor 11y ago

I like this rephrasing, but I think you only need n=2 subjects not n=3. To see why you should guess tails, set n=100.

CamperBob2 11y ago

The fact is, there aren't three volunteers, there's only one. Upon awakening, she has no information she didn't have when she went to sleep, which at first seems to be an unassailable argument in favor of the "halfers." To me, the problem with the "no new information" argument is that there's another popular paradox in which it also seems to apply, but fails spectacularly. In the Monty Hall paradox (https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem), you know in advance that the host is going to open a door at random and offer you a chance to change your initial guess. You're aware that the host knows what's behind the doors. You know that no matter what door you pick, one of the remaining ones has a goat behind it. And you know that the omniscient host will pick that door. So what new information do you gain when the host does what you knew he was going to do? One way to resolve the Monty Hall paradox is to rephrase it such that there are a million doors and the host opens all of the remaining ones except one. Of course it makes sense to switch doors in that scenario, since you knew that the host would avoid opening your initially-chosen door as well as the one with the car behind it. Your initial door has 1:1000000 odds of winning, while the only other closed door has odds of 1/2. So if you could find a way to transform the Sleeping Beauty problem into a Monty Hall problem, that could be a good way to settle the argument in favor of the "thirders." Or, if with some logical sleight-of-hand, you could show that SB is not isomorphic to MH, that could potentially be used to strengthen the no-new-information argument made by the "halfer" side. I'm leaning that way, because SB does not know upon awakening, and will not remember, whether the experiment will result in her being awakened twice. For all she knows, she's playing the same Monty Hall game for the second time. It's this lack of an unbroken timeline of conscious awareness that settles the question in favor of the halfers. To answer the thirders' rebuttal that SB would lose money if she were gambling on the coin toss with any odds other than 1:3, I'd say that's a completely different experiment. We aren't being asked to speculate on the set of events behind multiple awakenings, just one in isolation. The lack-of-new-information argument carries the day. Any arguments that involve repetition, clones, or gambling fail because they are responding to a different question. TL,DR: Don't gamble if the game involves erasing the players' memories. You will lose a lot of money in a hurry, even if your strategy is mathematically optimal.

URSpider94 11y ago

te_platt 11y ago

Your reframing of the question changes the problem. Let A be the probability of Heads (1/2) and B be the probability you are asked (1/2) then P(B|A) is 1/3. So the probability the coin came up heads is P(A|B) = P(B|A) * P(A)/P(B) = 1/3. Which is what you expected. In the original problem the probability of getting woken up is 1. So we end up with the probability of the coin being heads given that I was just woken up is (1 * 1/2) /1 = 1/2.

Retric 11y ago
Suppose I am going to flip a coin and give you money if you guess correctly. On heads I hand you 1$ on tails I hand you 1$ and add 1$ to your bank account without telling you. Now, after flipping the coin I ask you what the heads or tails? Clearly your going to pick tails. Why? Because it's the more rational choice based on the spoken rules. Going to sleep and losing memory is really just window dressing. The simple truth is before you went to sleep you should be able to say 1/3 odds of waking up to heads.
- timv 11y ago
  But that's still not the question. The stated question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads? Not, Should you _guess_ Heads or Tails? The probability of the coin toss _does not change_. The probability that "the reason they are waking you up is because the coin was Heads" does change, but that's not the question.
  - Retric 11y ago
    Clearly there is a 50:50 shot you get heads or tails from the coin. In your model of it's a 50:50 coin then you would have 50:25:25. The problem is waking up the second time does not reduce the odds of waking up the first time thus 50:50:50.
- te_platt 11y ago
  I think you changed the question. Before you go to sleep you know you have a 50% chance of waking up to heads and a 50% chance of waking up to tails twice. When you wake up you haven't learned anything so those probabilities can't have changed.

logicallee 11y ago

This is an incredibly powerful rephrasing, kudos. It's perfect, because for all intents and purposes you are "a[nother] person, who cannot tell if anyone [he] has been asked before", even though that other person is "also you." Very good! The half position is clearly, simply wrong.

mokus 11y ago

I think 1/2 argument makes essential use of the fact that the "2 people" are the same person at different times. Ultimately it comes down to a different idea of the way "being right twice" is valued. In the 1/3 position, the model is that each time you answer the question you get something good if you're right. In the 1/2 perspective, the idea is that after the whole experiment is over you either were simply correct or incorrect, and being correct twice doesn't count any extra.

timv 11y ago
You don't get to be "right twice" The question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads? It's not a guessing competition, it's asking you to assess the probability of a past event that you have no new information about.