Mirrors and prisoners
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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.
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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
- As said in another post, it reverses front and back, not left and right or up and down 2) since the answer is already decided, the actual probabilities are 1:1. What about the classics of... 1) which came first: The Chicken or the Egg? 2) If a tree falls in a forest, and no-one is there to hear it, does it make a sound? The answers are simple really.....
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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
2/3 = 1/3 + 1/2*2/3 There are 2 selections: Pick the first person to die, pick the second person to die The probability that any of the prisoners is the first person to die is 1/3. The probability that either of the remaining two prisoners is going to die is 1/2 times the probability that they are remaining. Which is 1/2*2/3.
This blanket smells like ham
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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:
each cell prisoner A thinks he has a 1-in-2 chance of surviving
Keyword here is 'thinks' - Prisoner A calculates the odds based on himself and prisoner C knowing that B will be shot, therefore 1 in 2 but the actual odds still include prisoner B even though he will be shot therefore real odds are 1 in 3. In the same way imagine flipping a coin, what are the odds of flipping heads twice in two flips? 1 in 4? If the first flip is heads then what are the odds? 1 in 2 or 1 in 4? Still 1 in 4! The problem hasn't changed only the perspective. You might say 1 in 2 because the first flip was heads but that would be changing the question, its the difference between:- What are the odds of flipping heads twice in two flips? and:- What are the odds of flipping heads twice in two flips, knowing the first will be heads? So, by the same arguement its the difference between:- What are the odds of prisoner A surviving? and What are the odds of prisoner A surviving knowing B will not? If you change the question the odds will change, but if you ask the same question then the odds remain 1 in 3. Hope that makes sense! :)
Apathy Rules - I suppose...
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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...
The mirror has rotated our image on its vertical axis. In other words, what you present to the mirror, it delivers the opposite back to you.
"A good athlete is the result of a good and worthy opponent." - David Crow
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
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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.
blackjack2150 wrote:
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?
Why not simply reference Let's Make a Deal, or the Monty Hall problem?
"A good athlete is the result of a good and worthy opponent." - David Crow
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
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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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Fred_Smith wrote:
each cell prisoner A thinks he has a 1-in-2 chance of surviving
Keyword here is 'thinks' - Prisoner A calculates the odds based on himself and prisoner C knowing that B will be shot, therefore 1 in 2 but the actual odds still include prisoner B even though he will be shot therefore real odds are 1 in 3. In the same way imagine flipping a coin, what are the odds of flipping heads twice in two flips? 1 in 4? If the first flip is heads then what are the odds? 1 in 2 or 1 in 4? Still 1 in 4! The problem hasn't changed only the perspective. You might say 1 in 2 because the first flip was heads but that would be changing the question, its the difference between:- What are the odds of flipping heads twice in two flips? and:- What are the odds of flipping heads twice in two flips, knowing the first will be heads? So, by the same arguement its the difference between:- What are the odds of prisoner A surviving? and What are the odds of prisoner A surviving knowing B will not? If you change the question the odds will change, but if you ask the same question then the odds remain 1 in 3. Hope that makes sense! :)
Apathy Rules - I suppose...
When I did A-level maths (don't know what yuor equivalent is - but it's the exam we take at 17/18 in order to (hopefully) get into university...) - we were given 8 questions on a 3-hour paper. Answering only 3 correctly would get you a pass, 4 a good pass. I answered 7 of them in a little over an hour, and had to sit there twiddling my thumbs for the remaining two (not allowed to leave early...) The 8-th question was a stats/probablility one; I didn't even bother trying. For some reason I have never been able to get my head around it - got a blank spot there. I like my maths to be absolute - not relative! Fred
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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...
The mirror has rotated our image on its vertical axis. In other words, what you present to the mirror, it delivers the opposite back to you.
"A good athlete is the result of a good and worthy opponent." - David Crow
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
DavidCrow wrote:
The mirror has rotated our image on its vertical axis
Exactly so. And why has it ignored the horizontal one?
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Fred_Smith wrote:
Why does a mirror reflection swap left-right but not up-down?
Mirrors do reflect up and down - lie on your side in front of one to prove it! ;P
Apathy Rules - I suppose...
Wouldn't it be easier to just hold the mirror sideways? :-D
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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
Statistics is the study of randomness. Since the guard is making the decision based on non-random criteria there is no chance that you will get shot. There is the absolute certainty which is unknown.
Need a C# Consultant? I'm available.
Happiness in intelligent people is the rarest thing I know. -- Ernest Hemingway -
DavidCrow wrote:
The mirror has rotated our image on its vertical axis
Exactly so. And why has it ignored the horizontal one?
Actually, a reflection is a half rotation. Two reflections can combine into either a translation or a rotation. Your mind is used to thinking in terms of a rotation not a reflection, that's why it gets confused.
This blanket smells like ham
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Statistics is the study of randomness. Since the guard is making the decision based on non-random criteria there is no chance that you will get shot. There is the absolute certainty which is unknown.
Need a C# Consultant? I'm available.
Happiness in intelligent people is the rarest thing I know. -- Ernest HemingwayStatistics works in both cases.
This blanket smells like ham
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Wouldn't it be easier to just hold the mirror sideways? :-D