Mirrors and prisoners
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The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
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The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
Fred_Smith wrote:
- The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both...
It doesn't swap left-right... :)
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Fred_Smith wrote:
- The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both...
It doesn't swap left-right... :)
alright - but it appears to. So why doesn't our brain make us think the image is upside down as well as reversed left-right?
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alright - but it appears to. So why doesn't our brain make us think the image is upside down as well as reversed left-right?
Fred_Smith wrote:
it appears to
I disagree - your right hand is on the right of the image and your left hand on the left. You only think it's reflected L-R because you interpret the image as being another person looking back at you, whose left hand is where your right hand's image is.
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The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
A variation of the second problem: You participate in a TV show where you have to choose one out one of three doors(A, B and C). One of them has a big prize - say a car - while tho other two have nothing. You choose for example door C. The presenter of the show opens door A and shows to you that it's empty. He gives you the chance to make up you mind and choose again between B and C. Should you swap to B or stick with the first choice, C? Answer: Switch to B. Even if it seems that there's a 50-50 chance for each, in fact the odds are 2-in-3 for B and 1-in-3 for C.
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A variation of the second problem: You participate in a TV show where you have to choose one out one of three doors(A, B and C). One of them has a big prize - say a car - while tho other two have nothing. You choose for example door C. The presenter of the show opens door A and shows to you that it's empty. He gives you the chance to make up you mind and choose again between B and C. Should you swap to B or stick with the first choice, C? Answer: Switch to B. Even if it seems that there's a 50-50 chance for each, in fact the odds are 2-in-3 for B and 1-in-3 for C.
You should have a fixed policy and you should swap. The swapper wins by not choosing the prize at first (odds 2/3) whereas the 'sticker' must get it right first time - odds 1/3.
Peter "Until the invention of the computer, the machine gun was the device that enabled humans to make the most mistakes in the smallest amount of time."
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The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
Fred_Smith wrote:
- The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both...
A curved "magnifier" mirror like a lens rotates an image as well, but a flat mirror simply umm.. mirrors it. It has to do with how light is reflected by each separate molecule. In a flat mirror they all reflect in the same directions, but in a curved one, a molecule at the center is angled differently than one at the outside edge. Roswell :)
"Angelinos -- excuse me. There will be civility today."
Antonio VillaRaigosa
City Mayor, Los Angeles, CA -
The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
About mirrors: Say "a mirror switches left and right" is wrong. It merely switches front and back, so that you are looking at your front when you are looking into a mirror. Basically, it's all about fraction and reflection. A mirror simply reflects. A lens also fractions the light, changing the "ray"'s direction and thus might change more than just front and back ;)
Cheers, Sebastian -- "If it was two men, the non-driver would have challenged the driver to simply crash through the gates. The macho image thing, you know." - Marc Clifton
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You should have a fixed policy and you should swap. The swapper wins by not choosing the prize at first (odds 2/3) whereas the 'sticker' must get it right first time - odds 1/3.
Peter "Until the invention of the computer, the machine gun was the device that enabled humans to make the most mistakes in the smallest amount of time."
So how come in a row of 100 cells, only 1-in-3 prisoners survive when they all have a 1-in-2 chance of doing so?
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The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
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So how come in a row of 100 cells, only 1-in-3 prisoners survive when they all have a 1-in-2 chance of doing so?
They don't have 1-in-2. They have 1-in-3 chances of survival. In your story, after the guardian says that B will go, the odds for survival are 1-in-3 for A and 2-in-3 for C.
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They don't have 1-in-2. They have 1-in-3 chances of survival. In your story, after the guardian says that B will go, the odds for survival are 1-in-3 for A and 2-in-3 for C.
blackjack2150 wrote:
after the guardian says that B will go, the odds for survival are 1-in-3 for A
How do you figure that? Once A knows that B will be shot, then he knows that either he or C will be spared = 1-in-2.
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blackjack2150 wrote:
after the guardian says that B will go, the odds for survival are 1-in-3 for A
How do you figure that? Once A knows that B will be shot, then he knows that either he or C will be spared = 1-in-2.
Here's the mistake in your logic. The one being shot is decided when all three of them are alive and from an external point of view all have 1-in-3 odds. So the odds are: A: 1/3 B or C: 2/3 (= 1/3 + 1/3) After A finds out that B will go down, the odds are A: 1/3 C: 2/3 because the odds of survival for B are "transferred" to C. It's not at all like flipping the coin after A and C are left. The one to be shot is known by the guard from the beginning.
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blackjack2150 wrote:
after the guardian says that B will go, the odds for survival are 1-in-3 for A
How do you figure that? Once A knows that B will be shot, then he knows that either he or C will be spared = 1-in-2.
Blackjack's explanation is nice and simple, here is another for those not so comfortable with probabilities: 3 prisoners each cell, labelled A, B, C 100 cells, 2 from each cell chosen to die, so 200 prisoners to die. At this stage all prisoners have equal 2/3 chance of dying. Label one prisoner from each cell 'A', take the 'A's from each cell giving a group of 100 prisoners, each with 2/3 chance of dying, on average 66 2/3 die From remaining 2 in each cell, at least one must die, remove one condemned man, call him 'B'. Being labelled a 'B' is a death sentence, all 100 'B's will die. Label the remaining man in each cell 'C'. In the remaining group of 100'C's, only 33 1/3 will die on average (33 1/3 = 200 - 100 - 66 2/3), so being labelled a 'C' means you only have a 1/3 chance of dying. Just as the knowledge used in labelling 'B' increases his chance of dying, so the knowledge used in labelling 'C' reduces his chance of dying.
Peter "Until the invention of the computer, the machine gun was the device that enabled humans to make the most mistakes in the smallest amount of time."
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The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
1. when you look straight into a mirror from 2 feet away, you see the world from the point of view of someone 2 feet on the other side of the mirror, looking out (at you). left and right aren't swapped; instead, the mirror world's Z axis is inverted compared to your own. what you see looking in, is what you would see looking out. when you see your own reflection, you see a person who (apparently) raises his left hand when you raise your right. but that's an illusion.
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Blackjack's explanation is nice and simple, here is another for those not so comfortable with probabilities: 3 prisoners each cell, labelled A, B, C 100 cells, 2 from each cell chosen to die, so 200 prisoners to die. At this stage all prisoners have equal 2/3 chance of dying. Label one prisoner from each cell 'A', take the 'A's from each cell giving a group of 100 prisoners, each with 2/3 chance of dying, on average 66 2/3 die From remaining 2 in each cell, at least one must die, remove one condemned man, call him 'B'. Being labelled a 'B' is a death sentence, all 100 'B's will die. Label the remaining man in each cell 'C'. In the remaining group of 100'C's, only 33 1/3 will die on average (33 1/3 = 200 - 100 - 66 2/3), so being labelled a 'C' means you only have a 1/3 chance of dying. Just as the knowledge used in labelling 'B' increases his chance of dying, so the knowledge used in labelling 'C' reduces his chance of dying.
Peter "Until the invention of the computer, the machine gun was the device that enabled humans to make the most mistakes in the smallest amount of time."
I am not convinced - this is all too clever by half: still seems to me that after asking the guard the question, A can sit out the rest of the night quite correct in his assumption that he has a 50-50 chance of survival. And all the A's can do this, but only 1/3rd of them will make it through dawn...
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I am not convinced - this is all too clever by half: still seems to me that after asking the guard the question, A can sit out the rest of the night quite correct in his assumption that he has a 50-50 chance of survival. And all the A's can do this, but only 1/3rd of them will make it through dawn...
If it helps them pass the night, I'm not going to correct them. It's not the sort of mistake you live to regret.
Peter "Until the invention of the computer, the machine gun was the device that enabled humans to make the most mistakes in the smallest amount of time."
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The recent thread on puzzling a 14-yr old got me thinking (again!) about a couple of classics that I still wonder about.. what's worst about them is that to this day I still can't make my mind up as to whether there's even a problem!! Sometimes I asy "oh come on it's obvious, don't be so dumb!" and others I say "hmm, yes, something's not adding up here..." 1) The real classic: Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both... 2) The 3-prisoners: There are 3 prisoners in a cell, A, B and C. The guard comes in one night and says he is going to execute two of them in the morning, but won't tell them who will be the lucky one, who he has chosen at random. So they're all sitting there thinking "I only have a 1-in-3 chance of surviving." When the guard comes back later, prisoner A whispers to him: "You're only sparing one of us, so whether or not I'm the lucky one, either B or C is going to die - please tell me the name of one of those two who you are going to shoot." the guard says "OK... B will be shot." So now A can sit there thinking "I know B dies, so now the other one must be either C or me - I have increased my chances of surviving to 1-in-2." But the guard had already made his mind up before being asked - so what are the real odds: 1-in-3 or 1-in-2? If that seems easy, consider this: rather than a single cell with 3 prisoners, imagine a whole cell block of 100 such cells, each with 3 prisoners, and the same thing going on in each of them... in each cell prisoner A thinks he has a 1-in-2 chance of surviving - but come an hour past dawn the next day, only 1-in-3 of them will still be alive.... cheers Fred
Fred_Smith wrote:
Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both...
Because they actually reverse front to back, see this explanation[^]
Graham Mirror, mirror, on the wall... The Wicked Queen
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Fred_Smith wrote:
Why does a mirror reflection swap left-right but not up-down? Symmetry would suggest they should do both...
Because they actually reverse front to back, see this explanation[^]
Graham Mirror, mirror, on the wall... The Wicked Queen
So it's even worse!! They reverse left-right, front-back... what have they got against up-down? Blatant discrimination, if you ask me: verticalism :-)
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Fred_Smith wrote:
it appears to
I disagree - your right hand is on the right of the image and your left hand on the left. You only think it's reflected L-R because you interpret the image as being another person looking back at you, whose left hand is where your right hand's image is.