Mirrors and prisoners
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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.
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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
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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...
A didn't ask the guard to name a random person of the 2 who would die; he asked for a random person who wasn't him and wasn't the survivor. If A had asked for a random person who wasn't the survivor, and the guard named B, then A would have a 50% chance of living. In theory, the guard could have said A would die, and since it didn't happen, A's chance would be 50%. In your original question, the odds did not change because A's question did not eliminate a purely random choice.
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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
I did A levels too - not maths though I didn't get it, strangely I went on to study Pure Maths at university! Statistics though isn't maths, its a black art practiced by wizards and magicians who follow the dark side. ;) Another thought - considering how difficult statistics is for most people to grasp its no suprise that it is a favorite of politicians and journalists, its a great tool for being able to make any point you like without ever being wrong!
Apathy Rules - I suppose...
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I did A levels too - not maths though I didn't get it, strangely I went on to study Pure Maths at university! Statistics though isn't maths, its a black art practiced by wizards and magicians who follow the dark side. ;) Another thought - considering how difficult statistics is for most people to grasp its no suprise that it is a favorite of politicians and journalists, its a great tool for being able to make any point you like without ever being wrong!
Apathy Rules - I suppose...
I dare say politicians and journalists don't have the faintest understanding of it either but, as you say, just use it to serve their ends. "Lies, damn lies, and statistics" as someone once said...
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Statistics works in both cases.
This blanket smells like ham
That inferential statistics requires randomness whereas descriptive statistics requires data to already exist.
Need a C# Consultant? I'm available.
Happiness in intelligent people is the rarest thing I know. -- Ernest Hemingway -
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.
My post just above is a link to your post, master. That one was far more succinct.
Cheers, Vıkram.
Be yourself, no matter what they say. - Sting, Englishman in New York.
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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
I don't see the problem here Fred. The size of the "Who Is Going to Die" sample is 3 (n=3) at the outset. A, B, and C all have a 1/3 chance of dying. However, B is taken out of the "Who is Going to Die" pool so now the sample size is 2 (n=2). Probability is based on possiible outcomes. You can't compare the proability percentage between scenarios where the possible outcomes are different. n=3 vs n-2. Probability examimes a possible outcome against all possible outcomes. If all possible outcomes change, then they can't be compared. So the probability is 1 out of 3 before chatting with guard amd 1 out of 2 after chatting with the guard.
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I don't see the problem here Fred. The size of the "Who Is Going to Die" sample is 3 (n=3) at the outset. A, B, and C all have a 1/3 chance of dying. However, B is taken out of the "Who is Going to Die" pool so now the sample size is 2 (n=2). Probability is based on possiible outcomes. You can't compare the proability percentage between scenarios where the possible outcomes are different. n=3 vs n-2. Probability examimes a possible outcome against all possible outcomes. If all possible outcomes change, then they can't be compared. So the probability is 1 out of 3 before chatting with guard amd 1 out of 2 after chatting with the guard.
Chadlling wrote:
So the probability is 1 out of 3 before chatting with guard amd 1 out of 2 after chatting with the guard.
So how come in the scenario of 100 such cells, only 1 out of 3 survive if they all have a 1 in 2 chance of doing so?
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Chadlling wrote:
So the probability is 1 out of 3 before chatting with guard amd 1 out of 2 after chatting with the guard.
So how come in the scenario of 100 such cells, only 1 out of 3 survive if they all have a 1 in 2 chance of doing so?
One in 2 of the A's and C's survive in each cell. None of the B's survive at all. Half of the A's and C's is 100/200, none of the B's is 0/100, for a net result of 100/300 surviving. Before the question you had: Person Chance of survival A 1/3 B 1/3 C 1/3 Total: 1 survivor per cell. After the question you have: Person Chance of survival A 1/2 B 0 C 1/2 Total: 1 survivor per cell.
-- You have to explain to them [VB coders] what you mean by "typed". their first response is likely to be something like, "Of course my code is typed. Do you think i magically project it onto the screen with the power of my mind?" --- John Simmons / outlaw programmer
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Chadlling wrote:
So the probability is 1 out of 3 before chatting with guard amd 1 out of 2 after chatting with the guard.
So how come in the scenario of 100 such cells, only 1 out of 3 survive if they all have a 1 in 2 chance of doing so?
Fred... multiplying the number of cells doesn't change the basic analysis. I think you are stretching the notion of "probability" outside its natural domain. Probability hinges on possible outcomes. What if the guard at 9:00 pm tells A, B, and C that 2 will die in the morning. but at midnight tells A he is going to be spared and tells B he is going to die and tells C nothing. Because probabilities hinge on possible outcomes, they will change for different people at different times. At 9:00 A/B/C all have a 1/3 chance of survival. At 12:00 the probability of survival is relevant to each person, dependent on how many possible outcomes are still viable. At midnight: A has a 100% probability of surviving B has a 0% chance of surviving C has a 33.3% chance of surviving You are thinking of probability as an absolute, when it is dependent on the individual, the timeline, and the availability of information. A and B know that C is going to die, but C doesn't, so his probablity (personal to him) is still 33.3%. If probability equals outcome/possible outcomes... there is no issue here. Because possible outcomes are entirely dependent on the flow of information. Let's say I pick one card out of a deck of cards, identify the card without showing it to you, and then putting it face down, while telling you that I will give you a million dollars if you tell me what card it is? What is the probability of winning the million? That is an unanswerable question, until you specify whose perspective you are talking about. Before you take your guess what is the probability that you will win. My perspective 1/52 Your perspective 1/52 God's perspective ?? There is no such thing as probability when talking about God because God operates outside of time, where there are no "possible outcomes" there is only THE outcome. You guess, but I haven't turned the card over. Your perspective still 1/52 My persoective ?? Probability no longer applies to me because the notion of possible outcomes has disapeared. Once I turn the card over, the notion of probability disappears for you too, since possible outcomes have also evaporated. It should not surprise you that probabilities change in accordance with information that changes the number of possible outcomes.
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- 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.....
If a woman is moaning and her husband isn't there to hear it, is it still his fault? Also eggs came first, eggs were around hundreds of millions of years before chickens.
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I did A levels too - not maths though I didn't get it, strangely I went on to study Pure Maths at university! Statistics though isn't maths, its a black art practiced by wizards and magicians who follow the dark side. ;) Another thought - considering how difficult statistics is for most people to grasp its no suprise that it is a favorite of politicians and journalists, its a great tool for being able to make any point you like without ever being wrong!
Apathy Rules - I suppose...
My son just got his GCSE results, Maths grade A, Further Maths grade A and Math and Stats grade C. We put the grade C down to Standard Deviation.
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If a woman is moaning and her husband isn't there to hear it, is it still his fault? Also eggs came first, eggs were around hundreds of millions of years before chickens.
it might not be his fault as such, but he still has some level of responsibility!