Sleeping beauty paradox
Poll
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30 members have voted
March 22nd, 2017 at 8:18:55 PM
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Sleeping Beauty volunteers to be the subject of an experiment. She is told truthfully that on Sunday she will be put to sleep. Immediately after, a fair coin will be flipped. Here is what will happen on happen, according to the flip:
Heads: She will be awakened on Monday only.
Tails: She will be awakened on Monday and Tuesday.
On each awakening she will be asked "What is the probability the coin landed on heads?" On any awakening, she will not know what day it is. After answering the question, she will be put back to sleep with an amnesia-inducing drug that will cause her to forget the awakening.
On Wednesday, she will wake up naturally.
Sleeping Beauty is a perfect logician.
How should she answer the question upon being awakened?
Heads: She will be awakened on Monday only.
Tails: She will be awakened on Monday and Tuesday.
On each awakening she will be asked "What is the probability the coin landed on heads?" On any awakening, she will not know what day it is. After answering the question, she will be put back to sleep with an amnesia-inducing drug that will cause her to forget the awakening.
On Wednesday, she will wake up naturally.
Sleeping Beauty is a perfect logician.
How should she answer the question upon being awakened?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 22nd, 2017 at 8:31:06 PM
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My gut feeling is that she should respond 2/3. But I can't wrap my head around the math.
Edit: Now I'm thinking 3/4. But the math still eludes me.
Edit: Now I'm thinking 3/4. But the math still eludes me.
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
March 22nd, 2017 at 10:15:34 PM
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Each time Sleeping Beauty is awakened she has no more information about the coin flip than she had before she went to sleep, because Sleeping Beauty will be awakened no matter whether the coin landed heads or tails, and she remains unaware of whether she is being awakened once or twice.
Given that (when awakened) she has no information that indicates anything about the outcome of the coin flip, she will answer "The probability that the coin landed on heads is 50%."
If the problem statement had required Sleeping Beauty to guess whether the coin had landed heads or tails upon any awakening with the proviso that she would be given one gold coin for any right answer, then she could maximize her EV by guessing "Tails" each time she is awakened. But, alas, that would be a different fairy tale.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
March 22nd, 2017 at 10:16:25 PM
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I'm tired and my first thoughts are usually wrong and calling it a paradox implies there may be multiple logically correct answers, but here's my shot:
Imagine running the experiment one million times. She'll likely be woken up a total of close to 1.5 million times, with 500,000 of the wake-ups coming from heads and 1,000,000 coming from tails, making the probability of heads 1/3
March 22nd, 2017 at 11:22:10 PM
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Is she asked the question on Wednesday too?
Simplicity is the ultimate sophistication - Leonardo da Vinci
March 22nd, 2017 at 11:33:04 PM
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She would reason that the probability of heads, P(H), is the sum of two numbers. The first number would be the product of the probability of its being Monday and the probability of heads given that it is Monday. The second number would be the product of the probability of its being Tuesday and the probability of heads given that it is Tuesday.
The first number is 2/3 times 1/2. The second number is 1/3 times 0. So, P(H) = 1/3.
The first number is 2/3 times 1/2. The second number is 1/3 times 0. So, P(H) = 1/3.
March 22nd, 2017 at 11:48:37 PM
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If today is Monday, 50%. If today is Tuesday, 0%.
RS.beerCount++;
March 23rd, 2017 at 12:03:14 AM
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1/4 - that's the probability.
March 23rd, 2017 at 3:58:00 AM
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Tails is going to produce the most days, but this is complicated by Monday being generated by both flips. She is not asked what day it is, but what flip it was. Plus she doesn't know what day it is! Since any flip is assumed to be 50-50 I think, she should just give a random answer, equal chance of heads or tails
the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: “Thanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious monster from Hell!” She is, after all, stone deaf. ... Arnold Snyder
March 23rd, 2017 at 4:49:02 AM
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Assuming Wednesday counts as an awakening, here is my plan. The coin will land on heads half the time. When it lands on heads half of the awakening days will be Monday, half Wednesday. The coin will land on tails half of the time. When it lands on tails 1/3 of the time the awakening will be Monday, 1/3 will be Tuesday, and 1/3 will be Wednesday.
So the Beauty will be awakened on a Monday 1/2 x 1/2 plus 1/2 x 1/3 = 5/12 of the time.
So the Beauty will be awakened on a Tuesday 0 x 1/2 plus 1/2 x 1/3 = 2/12 of the time.
So the Beauty will be awakened on a Wednesday 1/2 x 1/2 plus 1/2 x 1/3 = 5/12 of the time.
For the Monday awakenings, the probability of heads is 5/12 x 1 = 5/12
For the Tuesday awakenings the probability of heads is 2/12 x 0 = 0.
For the Wednesday awakenings the probability of heads is 5/12 x 1/2 = 5/24
Thus the probability of heads is 5/12 + 5/24 = 15/24.
If I'm wrong that is 10 minutes of my life I want back.
Edit- And if you don't count Wednesday as an awakening the question is too easy.
So the Beauty will be awakened on a Monday 1/2 x 1/2 plus 1/2 x 1/3 = 5/12 of the time.
So the Beauty will be awakened on a Tuesday 0 x 1/2 plus 1/2 x 1/3 = 2/12 of the time.
So the Beauty will be awakened on a Wednesday 1/2 x 1/2 plus 1/2 x 1/3 = 5/12 of the time.
For the Monday awakenings, the probability of heads is 5/12 x 1 = 5/12
For the Tuesday awakenings the probability of heads is 2/12 x 0 = 0.
For the Wednesday awakenings the probability of heads is 5/12 x 1/2 = 5/24
Thus the probability of heads is 5/12 + 5/24 = 15/24.
If I'm wrong that is 10 minutes of my life I want back.
Edit- And if you don't count Wednesday as an awakening the question is too easy.
March 23rd, 2017 at 4:55:30 AM
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Since the question is "What is the probability the coin landed on heads?" and isn't about what she should guess to match the result of the coin toss, the answer is 50%
March 23rd, 2017 at 6:06:54 AM
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Let me clarify that she is not asked on Wednesday.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 23rd, 2017 at 6:17:53 AM
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Assume you can set a probability p of it being Tuesday if the coin is tails.
There are three possible results:
(a) Heads, Monday has probability 1/2
(b) Tails, Monday has probability (1 - p) / 2
(c) Tails, Tuesday has probability p / 2
Regardless of the probability of it being Tuesday if the toss is tails, the probability of it being heads under the conditions is 1/2.
March 23rd, 2017 at 7:14:30 AM
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50%. I think it is a paradox to think anything else. The event hasn't yet taken place and some have said that Sleeping Beauty will say 2/3, for instance. This means that she has determined there to be a 2/3 chance of tails before she even goes to sleep on Sunday.
I heart Crystal Math.
March 23rd, 2017 at 7:14:49 AM
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Note: When I posted, and edited, I didn't notice that this is a poll.Quote: DJTeddyBearMy gut feeling is that she should respond 2/3. But I can't wrap my head around the math.
Edit: Now I'm thinking 3/4. But the math still eludes me.
Now that I see it, I vote 2/3. But I still can't provide any math. Just a knee-jerk gut feel.
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
March 23rd, 2017 at 8:02:48 AM
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Quote: WizardLet me clarify that she is not asked on Wednesday.
I hate you.... (not really)
So then I'm asking you, if she is asked on Wednesday.... am I correct?
March 23rd, 2017 at 8:12:21 AM
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Quote: TomGI'm tired and my first thoughts are usually wrong and calling it a paradox implies there may be multiple logically correct answers, but here's my shot:
Imagine running the experiment one million times. She'll likely be woken up a total of close to 1.5 million times, with 500,000 of the wake-ups coming from heads and 1,000,000 coming from tails, making the probability of heads 1/3
I agree. I don't see this as complicated. (which usually means I am missing something!)
March 23rd, 2017 at 8:13:28 AM
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Quote: WizardLet me clarify that she is not asked on Wednesday.
Here's my guess
She will flip a coin. If it is heads she will say 1/2, if it is tails, she will flip it again. If it is heads she will say 1/2 and if it is tails she will say 1/3.
Simplicity is the ultimate sophistication - Leonardo da Vinci
March 23rd, 2017 at 8:14:51 AM
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Quote: WizardSleeping Beauty volunteers to be the subject of an experiment. She is told truthfully that on Sunday she will be put to sleep. Immediately after, a fair coin will be flipped. Here is what will happen on happen, according to the flip:
Heads: She will be awakened on Monday only.
Tails: She will be awakened on Monday and Tuesday.
On each awakening she will be asked "What is the probability the coin landed on heads?" On any awakening, she will not know what day it is. After answering the question, she will be put back to sleep with an amnesia-inducing drug that will cause her to forget the awakening.
On Wednesday, she will wake up naturally.
Sleeping Beauty is a perfect logician.
How should she answer the question upon being awakened?
You didn't specifically state so, but SB is aware of the conditions of the experiment.... ? That heads results in one awakening over the next two days, and tails results in two awakenings?
March 23rd, 2017 at 8:34:56 AM
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I think that since this is a paradox it is fine to not use spoiler tags. There is not a definite answer. That said, let me argue my case for 1/3.
If this experiment were repeated thousands of times, then we could expect 1/3 of awakenings to be after a heads and 2/3 after a tails. Given that is an awakening, there is a 1/3 chance it was after a heads.
To test this, I did 10,000 flips. 4,915 were heads and 5,085 were tails.
So in total there were 4,915*1 + 5,085*2 = 15,085 awakenings. Of them, 4915 were after a heads flip. Thus the ratio of awakenings that were were after a heads was 4915/15085 = 32.58%, which is pretty close to 1/3.
If this experiment were repeated thousands of times, then we could expect 1/3 of awakenings to be after a heads and 2/3 after a tails. Given that is an awakening, there is a 1/3 chance it was after a heads.
To test this, I did 10,000 flips. 4,915 were heads and 5,085 were tails.
So in total there were 4,915*1 + 5,085*2 = 15,085 awakenings. Of them, 4915 were after a heads flip. Thus the ratio of awakenings that were were after a heads was 4915/15085 = 32.58%, which is pretty close to 1/3.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 23rd, 2017 at 10:05:32 AM
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I'm not sure I fully understand the question, so let me consider a different question in case it's essentially the same.
In a Mr & Mrs like game show there is a chance the final couple can win a prize. The host has two identical boxes and will place a prize in one of them. The husband is then asked a 50/50 question (and doesn't know whether his answer is correct or not). If the answer is correct a second prize is put in the other box. The wife picks a box (at random) and then...
(i) If the couple won the prize what were the chances the husband got the question correct?
(ii) Academically does it matter if there were orignally three boxes?
In a Mr & Mrs like game show there is a chance the final couple can win a prize. The host has two identical boxes and will place a prize in one of them. The husband is then asked a 50/50 question (and doesn't know whether his answer is correct or not). If the answer is correct a second prize is put in the other box. The wife picks a box (at random) and then...
(i) If the couple won the prize what were the chances the husband got the question correct?
(ii) Academically does it matter if there were orignally three boxes?
| Box 1 | Box 2 | |
|---|---|---|
| Correct | Prize | Prize |
| Wrong | Prize | - zip - |
March 23rd, 2017 at 11:24:43 AM
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Is she required to give the same answer on every wakening? Or can she change her answer as she gathers more information?
If she can change her answer then her strategy is: Always 1/2 and sometimes 0.
100% of the time she is woken on Monday and is effectively asked to guess the outcome of a fair coin flip that took place yesterday. So her answer would be 1/2.
If she is woken a second time she then knows with 100% certainty that 'tails' was the outcome on Sunday and will change her previous answer to 0.
If her answer must be fixed then her strategy is: 1/3, for the reasons others have already given.
If she can change her answer then her strategy is: Always 1/2 and sometimes 0.
100% of the time she is woken on Monday and is effectively asked to guess the outcome of a fair coin flip that took place yesterday. So her answer would be 1/2.
If she is woken a second time she then knows with 100% certainty that 'tails' was the outcome on Sunday and will change her previous answer to 0.
If her answer must be fixed then her strategy is: 1/3, for the reasons others have already given.
Views are my own...
March 23rd, 2017 at 1:48:43 PM
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Oh. come on folks. The problem statement does not ask Sleeping Beauty to guess the outcome of the coin flip (she doe not guess "Heads" or "Tails") and she has no incentive (as in a game show) to be correct on as many occasions (awakenings) as possible. It simply requires her to use logic and state the probability of the coin flip being Heads upon any awakening.
Before Sleeping Beauty went to sleep she knew that the probability of the random coin flip being Heads was 50%.
Upon any awakening, she has zero knowledge (either way) as to whether she has been awakened previously. She doesn't know the day of her awakening (I presume). All she knows is that she has been awakened.
If the flip was Heads, she was going to be awakened. If the flip was Tails, she was going to be awakened. Therefore, the mere fact that she has been awakened is equally consistent with both a Heads outcome and the Tails outcome. So, the act of awakening does not indicate or counter-indicate that the outcome was Heads.
Again: Before she went to sleep, she believed the probability of heads was 50% and now, upon awakening she has acquired no additional information that indicates the outcome of the coin flip . Logically, she has no evidence to change her expectation. The most logical answer she can give upon being awakened is 50%.
Before Sleeping Beauty went to sleep she knew that the probability of the random coin flip being Heads was 50%.
Upon any awakening, she has zero knowledge (either way) as to whether she has been awakened previously. She doesn't know the day of her awakening (I presume). All she knows is that she has been awakened.
If the flip was Heads, she was going to be awakened. If the flip was Tails, she was going to be awakened. Therefore, the mere fact that she has been awakened is equally consistent with both a Heads outcome and the Tails outcome. So, the act of awakening does not indicate or counter-indicate that the outcome was Heads.
Again: Before she went to sleep, she believed the probability of heads was 50% and now, upon awakening she has acquired no additional information that indicates the outcome of the coin flip . Logically, she has no evidence to change her expectation. The most logical answer she can give upon being awakened is 50%.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
March 23rd, 2017 at 3:22:03 PM
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This is a vague question - meaning what does "the probability" mean? As some have mentioned "probability" could be the "a priori" probability before the coin flip.Quote: WizardOn each awakening she will be asked "What is the probability the coin landed on heads?"
As gordonm888 stated, "How should she answer the question upon being awakened?" is also vague.
Upon awakening, the coin has already been flipped.
To an observer who knows the result of the coin flip, the current probability is P(H) = 0 or P(H) = 1.
Does the observer/awakener know the state of the coin flip, or which day it is?
The intended question could be "Sleeping Beauty, Do you think the coin flip landed heads? Please elaborate with your best guess of the current probability given only your knowledge." or "What do you think the 'a posteriori' likelihood is that the coin flip landed heads?" and the intended answers could be
(1) "I'm not sure, but to the best of my knowledge, P(H) = 1/3 or 1/2"
(2) Sleeping beauty should say "No. P(H)=0" 2/3 or 1/2 of the time, and "Yes. P(H)=1" 1/3 or 1/2 of the time.
(3) ....
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Whether 1/3 or 1/2 depends on how one amalgamates multiple answers. If each instance of Sleeping Beauty awakening is counted equally, then 1/3.
If each instance of coin flip is weighted equally (the two instances of Monday & Tuesday awakening only have 1/2 weight), then 1/2.
...this relates to the vagueness of "How should she answer the question upon being awakened?"
What is considered an instance of the experiment?
(a) each coin flip. Flip a coin 1,000 times. 1/2.
(b) each time sleeping beauty is asked. Flip a coin as many times as necessary so that we ask sleeping beauty 1,000 times. 1/3.
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It also assumes that Sleeping Beauty can not tell whether she has slept 1-2 days based on length of fingernails, how hungry she is, rumpledness of the hair, state of her period, how much she needs to urinate, internal time sense, etc...
Personally, if I wake after 12-18 hrs, rather than 6-8 hrs, I am much hungrier...
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IMO, the "perfect logician" might say "Your question is vague. Could you elaborate?"
...or the "perfect logician" might have figured out exactly what the question answerer wanted to hear ... and said that.
Last edited by: mamat on Mar 23, 2017
March 23rd, 2017 at 3:50:28 PM
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For one, I'm not really seeing the paradox. Also, the question isn't worded in a way to figure out what to optimize or the goal. It'd make more sense if it was worded like, "what probability should she guess such that her answer is closest to what actually happened?"
IE: If she guesses 1/3 (or 33.333%) probability it was heads, and it was heads, then she was off by 66.6667%. If it was tails, she was off by 33.3333%.
Or something like that.
As worded, I think the only answer (other than the one I gave on page 1) is 50%.
IE: If she guesses 1/3 (or 33.333%) probability it was heads, and it was heads, then she was off by 66.6667%. If it was tails, she was off by 33.3333%.
Or something like that.
As worded, I think the only answer (other than the one I gave on page 1) is 50%.
March 23rd, 2017 at 4:11:27 PM
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Memories of when I first studied Bayesian Networks in the 1990s:
This question reminds me of the difficulties in combining evidence from multiple sources/times.
It has similarities to the "Monty Hall problem" ... where there are unstated assumptions (which people see differently).
https://en.wikipedia.org/wiki/Monty_Hall_problem
Sun: Some evidence
Mon: Some more evidence.
Tue: Question asked.
Wed: More evidence.
Thu: Revised question asked.
Usually simple mathematical approaches will often double-count/triple-count evidence due the the "ad hoc" methods of calculating interdependencies in the evidence.
Idealistic Bayesian Network approaches have bottom-up and top-down sweeps (e.g. forward & backward in time) to propagate likelihoods and remove/minimize the double-counting or improper weighting. But then in real-life, approximations are often used because of complexity.
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Imagine a BJ game, where you have hole-carding info, shuffle tracking info, card counting, the actual cards seen in play so far, and the behavior of the other players. How does one actually combine all the info "correctly"?
In practice, we don't. It's too difficult for 99+% of humans.
Even if a human can maintain a 13 counts for all cards in the deck, the strategy modifications are too difficult.
But an AI system for diagnosing illnesses, engineering construction, etc... might want to be more accurate.
This question reminds me of the difficulties in combining evidence from multiple sources/times.
It has similarities to the "Monty Hall problem" ... where there are unstated assumptions (which people see differently).
https://en.wikipedia.org/wiki/Monty_Hall_problem
Sun: Some evidence
Mon: Some more evidence.
Tue: Question asked.
Wed: More evidence.
Thu: Revised question asked.
Usually simple mathematical approaches will often double-count/triple-count evidence due the the "ad hoc" methods of calculating interdependencies in the evidence.
Idealistic Bayesian Network approaches have bottom-up and top-down sweeps (e.g. forward & backward in time) to propagate likelihoods and remove/minimize the double-counting or improper weighting. But then in real-life, approximations are often used because of complexity.
-----
Imagine a BJ game, where you have hole-carding info, shuffle tracking info, card counting, the actual cards seen in play so far, and the behavior of the other players. How does one actually combine all the info "correctly"?
In practice, we don't. It's too difficult for 99+% of humans.
Even if a human can maintain a 13 counts for all cards in the deck, the strategy modifications are too difficult.
But an AI system for diagnosing illnesses, engineering construction, etc... might want to be more accurate.
March 23rd, 2017 at 8:12:00 PM
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Quote: mamat
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It also assumes that Sleeping Beauty can not tell whether she has slept 1-2 days based on length of fingernails, how hungry she is, rumpledness of the hair, state of her period, how much she needs to urinate, internal time sense, etc...
Personally, if I wake after 12-18 hrs, rather than 6-8 hrs, I am much hungrier...
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This is a clever lawyer-ly objection, but I feel I should point out that that the problem statement says:
"On any awakening, she will not know what day it is."
This is a problem much like Schrodinger's Cat which has a premise that an observer outside the box can receive no information about whether the Cat within the box has died. In this problem, an awakened Sleeping Beauty has no knowledge of what day it is and no knowledge of whether she has been awakened previously. Either accept the premise or move on.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
March 23rd, 2017 at 8:32:54 PM
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Quote: TheoHuxtableIs she required to give the same answer on every wakening? Or can she change her answer as she gathers more information?
She can answer whatever she wants at each awakening. However, she would remember nothing, so allowing changing of strategy doesn't help that I can see.
Quote: RSFor one, I'm not really seeing the paradox.
It is a paradox because there is no accepted answer.
Quote:Also, the question isn't worded in a way to figure out what to optimize or the goal. It'd make more sense if it was worded like, "what probability should she guess such that her answer is closest to what actually happened?"
Since what is being asked is a bit vague, let me offer a rewording, which the 1/2 camp may take issue with.
Let's say Sleeping Beauty is on a game show. They start by giving her $1,000,000. Then she agrees to go through this experiment 1,000 times. She is required to guess the same probability every awakening. A tally will be kept of the heads and tails leading to each awakening. At the end of the 1,000th experiment there should be close to 1,500 data points, because there are an average of 1.5 awakenings per experiment. The ratio of the "heads" data points to all of them will be calculated. Then it will be compared to Sleep Beauty's prediction. For every percentage point Sleep Beauty is off, $10,000 will be deducted from her million dollars. What probability should be render at every awakening?
Last edited by: Wizard on Mar 23, 2017
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 23rd, 2017 at 8:58:57 PM
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The math in these questions is always beyond me, but don't you mean Chuck Berry?
March 23rd, 2017 at 9:25:12 PM
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Humbly withdrawn; they died within three days of each other.
Carry on...
Carry on...
March 23rd, 2017 at 9:28:04 PM
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Quote: CalderThe math in these questions is always beyond me, but don't you mean Chuck Berry?
Nope. All due respect to Chuck Berry, but I identify with celebrities who hit their prime in the 70's or 80's. Chuck Barris had a huge mark on the 70's culture. I'd hate to calculate how much time I spent watching shows he starred or produced.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 23rd, 2017 at 9:55:49 PM
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There's 100% chance sleeping beauty will be woken up before wednsday.
The coin is only tossed 1 time, right after she is put to sleep. There's a 50% chance that a heads was tossed and she is woken up on Monday. And, a 50% chance a tails was tossed and she is awoken on Monday and Tuesday.
Sleeping beauty will not remember being awoken on Monday or Tuesday. She will think she slept straight through until Wednsday.
The coin is only flipped 1 time, so the results of the flips should be 500,000/500,000 in a million flips. Unless using a simulator, where it may be 550,000/450,000. But, ignoring that, the results would be 50/50 for the coin flips.
For the 1/3 people. The coin is only flipped 1 time, on Sunday night after sleeping beauty goes to sleep. In this case, the chance of being awoken one time is 50%, and the chance of being awoken 2 times IS ALSO 50%.
So, if there's 1.5 million awakenings, 33% being a heads, and 66% after a tails. Then the coin flip was 50/50. 50% heads and 50% tails.
So, the correct answer for sleeping beauty is 50%. Even if she woke up more often on tails, the coin flip results were 50/50.
The coin is only tossed 1 time, right after she is put to sleep. There's a 50% chance that a heads was tossed and she is woken up on Monday. And, a 50% chance a tails was tossed and she is awoken on Monday and Tuesday.
Sleeping beauty will not remember being awoken on Monday or Tuesday. She will think she slept straight through until Wednsday.
The coin is only flipped 1 time, so the results of the flips should be 500,000/500,000 in a million flips. Unless using a simulator, where it may be 550,000/450,000. But, ignoring that, the results would be 50/50 for the coin flips.
For the 1/3 people. The coin is only flipped 1 time, on Sunday night after sleeping beauty goes to sleep. In this case, the chance of being awoken one time is 50%, and the chance of being awoken 2 times IS ALSO 50%.
So, if there's 1.5 million awakenings, 33% being a heads, and 66% after a tails. Then the coin flip was 50/50. 50% heads and 50% tails.
So, the correct answer for sleeping beauty is 50%. Even if she woke up more often on tails, the coin flip results were 50/50.
March 24th, 2017 at 1:09:17 AM
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Quote: Wizard
Let's say Sleeping Beauty is on a game show. They start by giving her $1,000,000. Then she agrees to go through this experiment 1,000 times. She is required to guess the same probability every awakening. A tally will be kept of the heads and tails leading to each awakening. At the end of the 1,000th experiment there should be close to 1,500 data points, because there are an average of 1.5 awakenings per experiment. The ratio of the "heads" data points to all of them will be calculated. Then it will be compared to Sleep Beauty's prediction. For every percentage point Sleep Beauty is off, $10,000 will be deducted from her million dollars. What probability should be render at every awakening?
In this case, Sleeping Beauty, should answer 33% every time she is awoken. Because she will have awoken twice as often when a tails lands.
March 24th, 2017 at 4:04:08 AM
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Quote: WizardShe can answer whatever she wants at each awakening. However, she would remember nothing, so allowing changing of strategy doesn't help that I can see.
It is a paradox because there is no accepted answer.
Since what is being asked is a bit vague, let me offer a rewording, which the 1/2 camp may take issue with.
Let's say Sleeping Beauty is on a game show. They start by giving her $1,000,000. Then she agrees to go through this experiment 1,000 times. She is required to guess the same probability every awakening. A tally will be kept of the heads and tails leading to each awakening. At the end of the 1,000th experiment there should be close to 1,500 data points, because there are an average of 1.5 awakenings per experiment. The ratio of the "heads" data points to all of them will be calculated. Then it will be compared to Sleep Beauty's prediction. For every percentage point Sleep Beauty is off, $10,000 will be deducted from her million dollars. What probability should be render at every awakening?
Per the bold.
I don't know the definition of a paradox n stuff. But I thought a paradox had 2 (or more) answers where when you give one answer, that can't be correct because of some X, so the only appropriate answer is answer B. But answer B isn't correct because of Y, so only answer A is appropriate....but A isn't appropriate because of X......round and round it goes.
Or in other words -- there is more than 1 "correct" answer yet there is no correct answer. Or something like that.
Like the Indian who shoots an arrow at a tree at a steady rate of 10 ft/s. If he's 20 feet away, after 10 feet it'll take 1 second to travel. Traveling half the previous distance, it'll take 0.5 seconds to travel 5 more feet. (Now 15 feet @ 1.5 seconds) Then 0.25 seconds to travel 2.5 feet. (Now 17.5 feet @ 1.75 seconds) Then 0.125 seconds to travel 1.25 feet. Etc. The numbers never add up to a full 20 feet, even though they may get infinitely close to 20, won't reach 20, yet the arrow clearly hits the tree after 2 seconds.
March 24th, 2017 at 5:46:32 AM
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This is a different puzzle as all Sleeping Beauty has to do is estimate the ratio of Heads used to wake her up.Quote: Wizard...A tally will be kept of the heads and tails leading to each awakening...The ratio of the "heads" data points to all of them will be calculated. Then it will be compared to Sleep Beauty's prediction...
Since she can't change her answer she can give this at the beginning of the trial. She knows that, on average, there will be a similar number of heads (each Head creates one data point "H") and tails (each Tail creates two data points "T"). Thus on average there will be 1/3 "H" and 2/3 "T".
Another way of looking at it.
If every time they wake her up they either put an "H" ball in the bucket or a "T" ball in the bucket. Each game will either (Heads) result in one "H" ball or (Tails) two "T" balls in the bucket. For 1000 games there should be, on average, 500 "H" and 1000 "T", so the ratio is 1/3. This is what she should guess.
ps - I can't see the paradox with this.
March 24th, 2017 at 6:51:37 AM
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Quote: JyBrd0403Then the coin flip was 50/50. 50% heads and 50% tails.
The question isn't asking what the coin flip results were.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 24th, 2017 at 7:00:09 AM
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Quote: charliepatrickThis is a different puzzle as all Sleeping Beauty has to do is estimate the ratio of Heads used to wake her up.
Since she can't change her answer she can give this at the beginning of the trial. She knows that, on average, there will be a similar number of heads (each Head creates one data point "H") and tails (each Tail creates two data points "T"). Thus on average there will be 1/3 "H" and 2/3 "T".
The 1/2 camp evidently doesn't understand your argument, which I agree with. I was trying to put it another way, which may have only confused the issue more.
Quote:
Another way of looking at it.
If every time they wake her up they either put an "H" ball in the bucket or a "T" ball in the bucket. Each game will either (Heads) result in one "H" ball or (Tails) two "T" balls in the bucket. For 1000 games there should be, on average, 500 "H" and 1000 "T", so the ratio is 1/3. This is what she should guess.
You're preaching to choir with me.
Quote:ps - I can't see the paradox with this.
In the future, if I write about this, I'll call it a puzzle instead. Other sources refer to it as a "paradox" because it seems counter-intuitive that the answer is anything about 1/2. However, I don't see how the 1/2 side holds any water. Much like Alan's two-dice puzzle, just because so many people are wrong, doesn't mean the right answer should have any doubt.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 24th, 2017 at 8:08:11 AM
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I would consider that a substantial "rewording." Its as if you claim you are going to flip a fair coin numerous times but will count each "heads" double what you will count a "tails" when you tally the results. Then you act as if that changes what a "perfect logician" should predict as the probability of whether a toss is heads or tails.Quote: WizardSince what is being asked is a bit vague, let me offer a rewording, which the 1/2 camp may take issue with.
Did I miss something somewhere?
How's this for a rewording:
I will flip a fair coin numerous times. If a result is tails, I will record "1" in the tails column. If the result is heads, I will record a variable number in the heads column. That number will be biased, but you will have no information as to how I determine the bias. Your assignment is to determine which column will receive the higher total at the end of the flips. How do you come up with the answer if you are a "perfect logician"?
Seems like a wasted effort, not what I consider a paradox.
March 24th, 2017 at 8:42:44 AM
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It's a coin. The odds are 50/50. All the other stuff about sleeping and waking up and what day it is is just window dressing. The question was "What is the probability the coin landed on heads?" The answer to that question, given a theoretically perfectly fair coin, is always going to be 50/50.
Then again, I'm always wrong on these apparently simple odds questions!
EDIT: I didn't read the rewording or anyone else's answers. My answer relates to the OP.
Last edited by: TigerWu on Mar 24, 2017
March 24th, 2017 at 9:31:30 AM
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I suppose the most extreme rewording might be:
The girl (perfect logician) will be informed fully of the rules. She will then be placed in a trace, and a fair coin will be flipped one time. If it is heads, she will be executed in her sleep. If it is tails, she will be awakened and asked what the probability was for a heads result.
Under that wording, I think her answer would/should be: "The probability of heads was and is 50%. In this particular trial, the actual result appears to have been tails."
The girl (perfect logician) will be informed fully of the rules. She will then be placed in a trace, and a fair coin will be flipped one time. If it is heads, she will be executed in her sleep. If it is tails, she will be awakened and asked what the probability was for a heads result.
Under that wording, I think her answer would/should be: "The probability of heads was and is 50%. In this particular trial, the actual result appears to have been tails."
March 24th, 2017 at 9:59:43 AM
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I think a rewording is fair for this "paradox". 😂
I think a better way to describe or show 1/3 to be the correct answer is to say something like: "1/3 of the time when SB (sleeping beauty not super bowl, NFL) is awoken, the coin landed on heads."
I think a better way to describe or show 1/3 to be the correct answer is to say something like: "1/3 of the time when SB (sleeping beauty not super bowl, NFL) is awoken, the coin landed on heads."
March 24th, 2017 at 11:14:11 AM
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I think we had a similar discussion a few years ago. I don't recall the details, but I think it was something like a coin flip for which an observer sees the result is heads. Based on that and some other info, he is asked the probability that Player A had chosen heads.
My response was that the question did not make sense. The call of heads or tails was not a probabilistic event; it was a deterministic event, presumably determined by either Player A or his opponent.
Deterministic events, if repeated, may be seen after the fact to have a frequency distribution for a set of trials, but that does not mean there is an associated probability.
In the OP, the issue was described this way:
In my opinion, this is only meaningful as the probability of the 50-50 probabilistic event.
Edit: Yes, it is possible to analyze the likelihood that a deterministic event in the past went one way or another. ("Which candidate did the Wizard vote for in the presidential election? What is your confidence level in your answer?") That is different from analyzing the probability of a random event.
My response was that the question did not make sense. The call of heads or tails was not a probabilistic event; it was a deterministic event, presumably determined by either Player A or his opponent.
Deterministic events, if repeated, may be seen after the fact to have a frequency distribution for a set of trials, but that does not mean there is an associated probability.
In the OP, the issue was described this way:
Quote: WizardOn each awakening she will be asked "What is the probability the coin landed on heads?"
...
How should she answer the question upon being awakened?
In my opinion, this is only meaningful as the probability of the 50-50 probabilistic event.
Edit: Yes, it is possible to analyze the likelihood that a deterministic event in the past went one way or another. ("Which candidate did the Wizard vote for in the presidential election? What is your confidence level in your answer?") That is different from analyzing the probability of a random event.
March 24th, 2017 at 3:33:06 PM
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Quote: TigerWuIt's a coin. The odds are 50/50. All the other stuff about sleeping and waking up and what day it is is just window dressing. The question was "What is the probability the coin landed on heads?" The answer to that question, given a theoretically perfectly fair coin, is always going to be 50/50.
Then again, I'm always wrong on these apparently simple odds questions!
EDIT: I didn't read the rewording or anyone else's answers. My answer relates to the OP.
We can quit putting things in spoiler tags. To your post, again, the question is not whether or not the coin is fair. It is.
For those in the 1/2 camp, consider this question. I deal you a random card from a 52-card deck. I show it to 40 feet away. All you can tell is that it is a face card. What is the probability it is a Jack? I ask the same question to somebody 400 feet away who can't see it all. What is the probability according to him that it is a Jack?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 24th, 2017 at 3:36:24 PM
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Quote: DocI think we had a similar discussion a few years ago. I don't recall the details, but I think it was something like a coin flip for which an observer sees the result is heads. Based on that and some other info, he is asked the probability that Player A had chosen heads.
My response was that the question did not make sense. The call of heads or tails was not a probabilistic event; it was a deterministic event, presumably determined by either Player A or his opponent.
I think you're referring to the following puzzle, which had hundreds of posts to it. If I get a more than a few replies to this, I'll split it off.
Quote: Two coin puzzle
There are two coins in a bag. One has Heads on one side and Tails on the other. The other has Tails on both sides.
A coin is picked and random and one side exposed at random. That side is Tails. What is the probability the other side is Heads?
I think it is pretty much the same thing as the Sleeping Beauty problem but in different words.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 24th, 2017 at 10:35:51 PM
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Feel free to look for a logic mistake in the following. I would be surprised there is not one.
Identities:
(1) P(h and M): P(h) × P(M|h) = P(h|M) × P(M)
(2) P(h and T): P(h) × P(T|h) = P(h|T) × P(T)
(3) P(t and M): P(t) × P(M|t) = P(t|M) × P(M)
(4) P(t and T): P(t) × P(T|t) = P(t|T) × P(T)
In this puzzle:
The coin is either heads, h; or tails, t:
(5) P(h) + P(t) = 1
(6) P(h|M) + P(t|M) = 1
(7) P(h|T) + P(t|T) = 1
The day is either Monday, M; or Tuesday, T:
(8) P(M) + P(T) = 1
(9) P(M|h) + P(T|h) = 1
(10) P(M|t) + P(T|t) = 1
If the coin is heads, the awakening day must be Monday:
(11) P(M|h) = 1
(12) P(T|h) = 0
If an awakening day is Tuesday, the coin must be tails:
(13) P(h|T) = 0
(14) P(t|T) = 1
Then eqn.’s (1) – (4) become:
(15) P(h) = P(h|M) × P(M) from (1) and (11)
(16) 0=0 using (2), (12), and (13)
(17) P(t) × P(M|t) = P(t|M) × P(M) eqn. (3)
(18) P(t) × P(T|t) = P(T) from (4) and (14)
She knows that if an awakening day is Monday, then heads and tails were equally likely:
(19) P(h|M) = ½
(20) P(t|M) = ½
She knows that if the coin was tails, then it’s equally likely to be Monday or Tuesday:
(21) P(M|t) = ½
(22) P(T|t) = ½
Then:
(23) P(h) = ( ½ ) × P(M) from (15) and (19)
(24) P(t) × ( ½ ) = ( ½ ) × P(M) from (17), (21), and (20)
(25) P(t) × ( ½ ) = P(T) from (18) and (22)
And finally:
(26) P(h) = 1/3 from (23), (24), (25), (5), and (8)
Identities:
(1) P(h and M): P(h) × P(M|h) = P(h|M) × P(M)
(2) P(h and T): P(h) × P(T|h) = P(h|T) × P(T)
(3) P(t and M): P(t) × P(M|t) = P(t|M) × P(M)
(4) P(t and T): P(t) × P(T|t) = P(t|T) × P(T)
In this puzzle:
The coin is either heads, h; or tails, t:
(5) P(h) + P(t) = 1
(6) P(h|M) + P(t|M) = 1
(7) P(h|T) + P(t|T) = 1
The day is either Monday, M; or Tuesday, T:
(8) P(M) + P(T) = 1
(9) P(M|h) + P(T|h) = 1
(10) P(M|t) + P(T|t) = 1
If the coin is heads, the awakening day must be Monday:
(11) P(M|h) = 1
(12) P(T|h) = 0
If an awakening day is Tuesday, the coin must be tails:
(13) P(h|T) = 0
(14) P(t|T) = 1
Then eqn.’s (1) – (4) become:
(15) P(h) = P(h|M) × P(M) from (1) and (11)
(16) 0=0 using (2), (12), and (13)
(17) P(t) × P(M|t) = P(t|M) × P(M) eqn. (3)
(18) P(t) × P(T|t) = P(T) from (4) and (14)
She knows that if an awakening day is Monday, then heads and tails were equally likely:
(19) P(h|M) = ½
(20) P(t|M) = ½
She knows that if the coin was tails, then it’s equally likely to be Monday or Tuesday:
(21) P(M|t) = ½
(22) P(T|t) = ½
Then:
(23) P(h) = ( ½ ) × P(M) from (15) and (19)
(24) P(t) × ( ½ ) = ( ½ ) × P(M) from (17), (21), and (20)
(25) P(t) × ( ½ ) = P(T) from (18) and (22)
And finally:
(26) P(h) = 1/3 from (23), (24), (25), (5), and (8)
Last edited by: ChesterDog on Mar 24, 2017
March 24th, 2017 at 11:08:43 PM
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Your "Two coin puzzle" is not as ambiguously worded as the "paradox."Quote: WizardI think you're referring to the following puzzle, which had hundreds of posts to it. If I get a more than a few replies to this, I'll split it off.
I think it is pretty much the same thing as the Sleeping Beauty problem but in different words.Quote: Two coin puzzle
There are two coins in a bag. One has Heads on one side and Tails on the other. The other has Tails on both sides.
A coin is picked and random and one side exposed at random. That side is Tails. What is the probability the other side is Heads?
What makes a lot of "paradox"es interesting is
(1) English is ambiguous
(2) People can make different unstated assumptions, and thus vehemently believe that they have the "right" answer, and that others are "wrong".
It's like a visual illusion, or verbal joke. Some people "get it", some don't.
When no one get's it, you look really boring & stupid, because no one is interested.
March 25th, 2017 at 8:48:21 AM
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Quote: Wizard
For those in the 1/2 camp, consider this question. I deal you a random card from a 52-card deck. I show it to 40 feet away. All you can tell is that it is a face card. What is the probability it is a Jack?
I argue that there are two difference questions being conflated here:
1) What are the odds of being dealt a jack from a deck of cards? 4 Jacks out of 52 cards = 1/13
2) What are the odds of any random face card being a jack? 4 Jacks out of 12 face cards = 1/3
Quote:I ask the same question to somebody 400 feet away who can't see it all. What is the probability according to him that it is a Jack?
He would say 1/13. In effect we have the same answer as far as a Jack being drawn from a deck of cards. You are just giving me more information with regards to a separate question.
March 25th, 2017 at 9:08:31 AM
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Quote: TigerWuI argue that there are two difference questions being conflated here:
There are. I'm trying to make the point that the same probability question can have two different answers according to who is being asked.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 25th, 2017 at 9:57:20 AM
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Quote: WizardThere are. I'm trying to make the point that the same probability question can have two different answers according to who is being asked.
I think it is more accurate to say "two different answers according to the information available to the one being asked." In the card example, the difference is the fact that one observer has more info than the other. In the SB example, she has the exact same info every time she has to answer... In fact, if she was to answer the question before being put to sleep the first time, what would her answer be? This may be the "pair of ducks".
Simplicity is the ultimate sophistication - Leonardo da Vinci
March 25th, 2017 at 9:58:10 AM
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Quote: WizardThere are. I'm trying to make the point that the same probability question can have two different answers according to who is being asked.
Just spent more time than I should have trying to figure out if you should have used "whom" instead of "who". :( I'm pretty sure it's "whom", but the Internet can't verify that for me. It has failed me once again.
Believe it or not, EvenBob wrote of a paradox sort of like this / related. Don't remember how it was worded, but something like this: You're a roulette player and your friend can tell the future. He tells you which column to bet on, such that you have a 50% chance to win every spin (true probability is 12/38 or about 1/3). Sometimes he tells you which is the winner correctly and other times he doesn't, as winning every bet would be suspicious. Now, there are 2 others at the tables, with the same deal with your friend.
If each player has a bet on a different column, do they each have a 33% chance to win or 50% chance?
I think this was EvenBob's "proof" the likelihood of something happening could exceed 100%, or something like that.
