Poll: A little math problem

[Dboyz-x.etown]

...Is English your first language?
___________
so if "the other puppy" "fulfilled the "at least 1 is male" requirement," then the answer to the question is 100%
___________

--If, by answer to the question you mean "probability for that particular scenario", then
yes. However, that is not the only scenario which can exist, so you have to account for the rest as well. The scenario you described is analogous to either Universe 1 or 2 in my example.
- - - - - - - - - - - - - - - - - - - -
___________
remember, we're not being asked the probability that both puppies are male, but rather, "What is the probability that the other one is a male?"
___________
--Exactly.

 

[Samirat]

Nah, I like my quantum mechanics answer of it being more or less pointless to try to determine, as it could be the same dog occupying multiple locations at once. Though that then degrades into there might be only one dog, and the others are merely quantum madness, or there might be an infinite number of dogs occupying the same point in space-time.

Yeah, sure, that would be cool, if it actually worked. Fortunately, quantum mechanics has no place in the real world.

Keep telling yourself that... but the effects of quantum mechanics are actually all around us...

Edit: is this then the 1000th post in the thread?
Edit 2: you realise that quantum mechanics arises from statistics? The idea of MF being equivalent to FM relies on a 'superposition of states' much like Schroedinger's cat. We know that one dog is male, but not which one.

The key word here is quantum. Quantum mechanics is only relevant to mechanical systems close to the atomic scale. The Schroedinger's Cat problem occurs in a very different world from ours, and the situation is impossible to replicate in real life. It relies on a complete uncertainty that is only theoretically possible on such a large scale.

And probability such as this doesn't rely on "superposition of states." Superposition of states is the quantum mechanical explanation for probability. In no important way does Probability Theory rely on quantum mechanics. Superposition of states is merely an explanation for probability from the quantum mechanical perspective.

 

[Lukeje]

Nah, I like my quantum mechanics answer of it being more or less pointless to try to determine, as it could be the same dog occupying multiple locations at once. Though that then degrades into there might be only one dog, and the others are merely quantum madness, or there might be an infinite number of dogs occupying the same point in space-time.

Yeah, sure, that would be cool, if it actually worked. Fortunately, quantum mechanics has no place in the real world.

Keep telling yourself that... but the effects of quantum mechanics are actually all around us...

Edit: is this then the 1000th post in the thread?
Edit 2: you realise that quantum mechanics arises from statistics? The idea of MF being equivalent to FM relies on a 'superposition of states' much like Schroedinger's cat. We know that one dog is male, but not which one.

The key word here is quantum. Quantum mechanics is only relevant to mechanical systems close to the atomic scale. The Schroedinger's Cat problem occurs in a very different world from ours, and the situation is impossible to replicate in real life. It relies on a complete uncertainty that is only theoretically possible on such a large scale.

And probability such as this doesn't rely on "superposition of states." Superposition of states is the quantum mechanical explanation for probability. In no important way does Probability Theory rely on quantum mechanics. Superposition of states is merely an explanation for probability from the quantum mechanical perspective.

I stated that quantum mechanics depends upon statistics, not that statistics depends upon quantum mechanics. 'Superposition of states' is merely the terminology that I am most familiar with. My viewpoint can be envisaged as so: if we were plotting a graph of the states, we would have MM on its own, with a distinct probability (1/3). We would then have an MF and an FM term. They would occupy the same space on the graph and 'superpose' to form a probability equal to the sum of the two probabilities (2/3).
The best idea would indeed be (as Dboyz says) to set up an n-dimensional space where all possibilities play out, and then work out the probabilities of each, but alas, this is not practicable.
With regard to quantum effects being unobservable to us at this level, quantum computers have come a long way from their initial theoretical conception, and may soon (i.e. in 20 yrs or so) be useful in every day life.

 

[Samirat]

Nah, I like my quantum mechanics answer of it being more or less pointless to try to determine, as it could be the same dog occupying multiple locations at once. Though that then degrades into there might be only one dog, and the others are merely quantum madness, or there might be an infinite number of dogs occupying the same point in space-time.

Yeah, sure, that would be cool, if it actually worked. Fortunately, quantum mechanics has no place in the real world.

Keep telling yourself that... but the effects of quantum mechanics are actually all around us...

Edit: is this then the 1000th post in the thread?
Edit 2: you realise that quantum mechanics arises from statistics? The idea of MF being equivalent to FM relies on a 'superposition of states' much like Schroedinger's cat. We know that one dog is male, but not which one.

The key word here is quantum. Quantum mechanics is only relevant to mechanical systems close to the atomic scale. The Schroedinger's Cat problem occurs in a very different world from ours, and the situation is impossible to replicate in real life. It relies on a complete uncertainty that is only theoretically possible on such a large scale.

And probability such as this doesn't rely on "superposition of states." Superposition of states is the quantum mechanical explanation for probability. In no important way does Probability Theory rely on quantum mechanics. Superposition of states is merely an explanation for probability from the quantum mechanical perspective.

I stated that quantum mechanics depends upon statistics, not that statistics depends upon quantum mechanics. 'Superposition of states' is merely the terminology that I am most familiar with. My viewpoint can be envisaged as so: if we were plotting a graph of the states, we would have MM on its own, with a distinct probability (1/3). We would then have an MF and an FM term. They would occupy the same space on the graph and 'superpose' to form a probability equal to the sum of the two probabilities (2/3).
The best idea would indeed be (as Dboyz says) to set up an n-dimensional space where all possibilities play out, and then work out the probabilities of each, but alas, this is not practicable.
With regard to quantum effects being unobservable to us at this level, quantum computers have come a long way from their initial theoretical conception, and may soon (i.e. in 20 yrs or so) be useful in every day life.

Hmm, I thought when you said that the "idea of MF being equivalent to FM relies on a 'superposition of states,'" you meant that probability depended on quantum mechanics. I'm just saying, it doesn't "rely" on quantum mechanics. It can be explained by ideas of quantum mechanics, but Probability Theory is much older and more fleshed out. We are setting up a space where all possibilities play out. It's this thing:

MM
MF
FM

 

[Lukeje]

Hmm, I thought when you said that the "idea of MF being equivalent to FM relies on a 'superposition of states,'" you meant that probability depended on quantum mechanics. I'm just saying, it doesn't "rely" on quantum mechanics. It can be explained by ideas of quantum mechanics, but Probability Theory is much older and more fleshed out. We are setting up a space where all possibilities play out. It's this thing:

MM
MF
FM

Hey, its just 'cos I've never formally studied statistics... I just catch glimpses of it through this window known as quantum mechanics...

 

[Murrah]

According to a site that i've been, it is apparently a fact that there are more population of male in the world than females. So im guessing that the probability of the other is a male is 66%.

 

[geizr]

Nah, I like my quantum mechanics answer of it being more or less pointless to try to determine, as it could be the same dog occupying multiple locations at once. Though that then degrades into there might be only one dog, and the others are merely quantum madness, or there might be an infinite number of dogs occupying the same point in space-time.

Yeah, sure, that would be cool, if it actually worked. Fortunately, quantum mechanics has no place in the real world.

Keep telling yourself that... but the effects of quantum mechanics are actually all around us...

Edit: is this then the 1000th post in the thread?
Edit 2: you realise that quantum mechanics arises from statistics? The idea of MF being equivalent to FM relies on a 'superposition of states' much like Schroedinger's cat. We know that one dog is male, but not which one.

The key word here is quantum. Quantum mechanics is only relevant to mechanical systems close to the atomic scale. The Schroedinger's Cat problem occurs in a very different world from ours, and the situation is impossible to replicate in real life. It relies on a complete uncertainty that is only theoretically possible on such a large scale.

And probability such as this doesn't rely on "superposition of states." Superposition of states is the quantum mechanical explanation for probability. In no important way does Probability Theory rely on quantum mechanics. Superposition of states is merely an explanation for probability from the quantum mechanical perspective.

Wow, I can't believe this mess is still going on, but, speaking as a physics student, I feel I have to respond to this.

Quantum mechanics is, in reality, relevant at all levels, not just the small world of atomic and sub-atomic particles, just as Relativity is relevant on all scales, not just the large and fast. What happens is that the deviations from the classical theories for most "everyday" phenomena are usually below our ability to measure. Even so, there are many macroscopic phenomena that occur visibly, even on our classical/everyday level, that require the use of quantum mechanics to predict and understand(a couple of examples off the top are super-fluids and super-conductors).

Superposition is not just an explanation for probability in quantum mechanics; it's a very real occurrence. The easiest example to see superposition in action is the Stern-Gerlach experiment(look it up; I'm not going to detail it here).

However, there is an easier way to see superposition at the everyday level, and all it will cost is the price of admission to a 3D movie. The next time you go to a 3D movie, take 3 pairs of the polarized glasses they give you(just use the ones your friends or family get, cause I doubt they will actually give you extras). Take two of the glasses and hold the left-eye lenses up to each other so you can look through both. The rotate one of the lenses until you can't see all the way through anymore. At that point, the lenses should be oriented 90-degrees to each other. Now, take the third pair of glasses and insert the left lens of it between the left lenses of the other two. Then rotate the third glasses until its left lens is at a 45-degree angle to both of the first two lenses you used. You'll see something really weird happen: you're able to see through all three lenses! This is superposition in action.

The explanation is this: the lens of the 3D glasses polarize the light either horizontally or vertically(the left lens will polarize one way while the right lens polarizes the other way). Let's say, for the sake of explanation, the left lens polarizes vertically. So, with the first lens held normally, only lets through light that is vertically polarized. Normal light all around us has all kinds of directions of polarization and even has strange polarizations called circular polarizations(but these aren't important in this case). When the light passes through the first lens, all polarizations except the vertical polarizations are filtered out; so only the vertical polarizations make it through. When you take the second set of glasses and turn its left lens 90-degrees to the first, you make a horizontal polarization filter that only lets through horizontally polarized light. Well, the light from the first lens is all vertical, no horizontal components. So, no light gets through the second lens.

Now, here's where superposition comes into play. The third lens is at a 45-degree angle. So, it only lets through let that is has a 45-degree polarization. But, 45-degree polarization is composed of equal parts of vertical and horizontal polarization. Equivalently, you can say that vertical and horizontal polarizations are composed of equal parts of 45-degree polarizations(which include 135, 225, and 315 degree polarizations). Well, the vertical polarization from the first lens can be looked upon as being composed of equal parts of 45-degree and 135-degree polarizations. Well, the third lens lets through 45-degree polarizations, so half the light of the first lens gets through the third. By extension, you can then see that half the light from the third lens is able to make it through the second lens. Light makes it all the way through the three lens system and not the two lens system because of superposition of states(which are the polarizations).

In short, superposition of states is not a probabilistic view, it's a compositional view. The object actually is existing in multiple states simultaneously until you act upon it such to single at a particular state or set of states. In more technical language, we say that acting upon the wave-function with an observable operator collapses the wave-function to an eigenstate of that operator. In the example above, the observable operators were each of the polarized lenses from our 3D glasses, the eigenstates were the different polarizations, and the wave-function is the light itself. Notice that which eigenstates(polarizations) we used depended on the operator(lens) we used.

Now, in a real quantum system, you get things like decoherence and state evolution that may further change the mix of eigenstates in the wavefunction, even though a specific eigenstate may have been selected at an earlier time.

Sorry for the windy response, but I just felt the need to clear up that little bit of misunderstanding.

 

[Samirat]

Nah, I like my quantum mechanics answer of it being more or less pointless to try to determine, as it could be the same dog occupying multiple locations at once. Though that then degrades into there might be only one dog, and the others are merely quantum madness, or there might be an infinite number of dogs occupying the same point in space-time.

Yeah, sure, that would be cool, if it actually worked. Fortunately, quantum mechanics has no place in the real world.

Keep telling yourself that... but the effects of quantum mechanics are actually all around us...

Edit: is this then the 1000th post in the thread?
Edit 2: you realise that quantum mechanics arises from statistics? The idea of MF being equivalent to FM relies on a 'superposition of states' much like Schroedinger's cat. We know that one dog is male, but not which one.

The key word here is quantum. Quantum mechanics is only relevant to mechanical systems close to the atomic scale. The Schroedinger's Cat problem occurs in a very different world from ours, and the situation is impossible to replicate in real life. It relies on a complete uncertainty that is only theoretically possible on such a large scale.

And probability such as this doesn't rely on "superposition of states." Superposition of states is the quantum mechanical explanation for probability. In no important way does Probability Theory rely on quantum mechanics. Superposition of states is merely an explanation for probability from the quantum mechanical perspective.

Wow, I can't believe this mess is still going on, but, speaking as a physics student, I feel I have to respond to this.

Quantum mechanics is, in reality, relevant at all levels, not just the small world of atomic and sub-atomic particles, just as Relativity is relevant on all scales, not just the large and fast. What happens is that the deviations from the classical theories for most "everyday" phenomena are usually below our ability to measure. Even so, there are many macroscopic phenomena that occur visibly, even on our classical/everyday level, that require the use of quantum mechanics to predict and understand(a couple of examples off the top are super-fluids and super-conductors).

Superposition is not just an explanation for probability in quantum mechanics; it's a very real occurrence. The easiest example to see superposition in action is the Stern-Gerlach experiment(look it up; I'm not going to detail it here).

However, there is an easier way to see superposition at the everyday level, and all it will cost is the price of admission to a 3D movie. The next time you go to a 3D movie, take 3 pairs of the polarized glasses they give you(just use the ones your friends or family get, cause I doubt they will actually give you extras). Take two of the glasses and hold the left-eye lenses up to each other so you can look through both. The rotate one of the lenses until you can't see all the way through anymore. At that point, the lenses should be oriented 90-degrees to each other. Now, take the third pair of glasses and insert the left lens of it between the left lenses of the other two. Then rotate the third glasses until its left lens is at a 45-degree angle to both of the first two lenses you used. You'll see something really weird happen: you're able to see through all three lenses! This is superposition in action.

The explanation is this: the lens of the 3D glasses polarize the light either horizontally or vertically(the left lens will polarize one way while the right lens polarizes the other way). Let's say, for the sake of explanation, the left lens polarizes vertically. So, with the first lens held normally, only lets through light that is vertically polarized. Normal light all around us has all kinds of directions of polarization and even has strange polarizations called circular polarizations(but these aren't important in this case). When the light passes through the first lens, all polarizations except the vertical polarizations are filtered out; so only the vertical polarizations make it through. When you take the second set of glasses and turn its left lens 90-degrees to the first, you make a horizontal polarization filter that only lets through horizontally polarized light. Well, the light from the first lens is all vertical, no horizontal components. So, no light gets through the second lens.

Now, here's where superposition comes into play. The third lens is at a 45-degree angle. So, it only lets through let that is has a 45-degree polarization. But, 45-degree polarization is composed of equal parts of vertical and horizontal polarization. Equivalently, you can say that vertical and horizontal polarizations are composed of equal parts of 45-degree polarizations(which include 135, 225, and 315 degree polarizations). Well, the vertical polarization from the first lens can be looked upon as being composed of equal parts of 45-degree and 135-degree polarizations. Well, the third lens lets through 45-degree polarizations, so half the light of the first lens gets through the third. By extension, you can then see that half the light from the third lens is able to make it through the second lens. Light makes it all the way through the three lens system and not the two lens system because of superposition of states(which are the polarizations).

In short, superposition of states is not a probabilistic view, it's a compositional view. The object actually is existing in multiple states simultaneously until you act upon it such to single at a particular state or set of states. In more technical language, we say that acting upon the wave-function with an observable operator collapses the wave-function to an eigenstate of that operator. In the example above, the observable operators were each of the polarized lenses from our 3D glasses, the eigenstates were the different polarizations, and the wave-function is the light itself. Notice that which eigenstates(polarizations) we used depended on the operator(lens) we used.

Now, in a real quantum system, you get things like decoherence and state evolution that may further change the mix of eigenstates in the wavefunction, even though a specific eigenstate may have been selected at an earlier time.

Sorry for the windy response, but I just felt the need to clear up that little bit of misunderstanding.

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

 

[Geamo]

Bloody hell..is this still going?

50%. The previous outcome has no effect on the current in this case.

"Random loot is random."

 

[Lukeje]

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

Light does some freaky ass things that seem to defy the principle of sufficient causation itself.

Its... magic! (Or just a consequence of photons being particles travelling in waves... or waves that have the properties of particles... or both until you try to measure a property only strictly defined for one of the two...)

 

[geizr]

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

Yes, it is quite true. I've done it myself many times(it's one of the things I like wowing people with when I am at 3D movies). If you have polarizing filters, you can use those. It's the same thing because that's what the 3D glasses are.

This is the prime example quantum mechanics instructors use to demonstrate the Stern-Gerlach experiment to first year graduate students in physics.

 

[geizr]

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

Light does some freaky ass things that seem to defy the principle of sufficient causation itself.

I have no idea what this principle of sufficient causation is that you mention, but this is simple physics that is easily calculable by any junior year undergrad or first year graduate physics student. Nothing freaky about it. In fact, dealing with light is the easy stuff.

Now if you want freaky, try dealing with identical particle mixed states. There you find that the particles suffer an identity crisis. Particle1 actually will not be able to decide if it is particle1 or particle2, and particle2 will have the same trouble.

 

[geizr]

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

Light does some freaky ass things that seem to defy the principle of sufficient causation itself.

Its... magic! (Or just a consequence of photons being particles travelling in waves... or waves that have the properties of particles... or both until you try to measure a property only strictly defined for one of the two...)

In this case with the polarizations, we are using the wave nature of the light. Explaining the truth of the wave-particle duality of nature is...hard. Really, really, freaking hard. And even harder to make real sense of it. You begin to understand it in a back-of-the-mind intuition, but it's impossible to really convey the true reality of it to anyone who is not well experienced with the math and machinery of quantum mechanics(I tried explaining eigenstates and eigenvalues to a friend of mine once. Got about half-way before he asked me to just stop). It seems easy at first to just say that it means things act like a particle and a wave. But, it's not until you start really truly thinking about it that you discover you don't really know what that truly means. We just know how to calculate it.

 

[geizr]

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

Oh, just to be clear, it is critical that the 45-degree filter be placed between the vertical and horizontal filters, else it'll all just remain opaque.

 

[Lukeje]

In this case with the polarizations, we are using the wave nature of the light. Explaining the truth of the wave-particle duality of nature is...hard. Really, really, freaking hard. And even harder to make real sense of it. You begin to understand it in a back-of-the-mind intuition, but it's impossible to really convey the true reality of it to anyone who is not well experienced with the math and machinery of quantum mechanics(I tried explaining eigenstates and eigenvalues to a friend of mine once. Got about half-way before he asked me to just stop). It seems easy at first to just say that it means things act like a particle and a wave. But, it's not until you start really truly thinking about it that you discover you don't really know what that truly means. We just know how to calculate it.

Its easier to think about it in an abstract way (i.e. via linear algebra [eigenstates/eigenfunctions]) than it actually is to conceptualise it.

 

[beddo]

I was looking around on the internet, and I found something about Marilyn vos Savant. When I was reading about it I saw an interesting math problem. This was taken directly from Wikipedia, so I'm not certain about its accuracy in wording:

A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other one is a male?

Please answer the poll before reading on.

I think Vos Savant said that it was a 1/3 chance, although the Wikipedia article wasn't very clear about it. However, I though it was interesting, and since this is seems to be a smart forum, I thought I'd ask it here.

This is just probablity. These questions are often counter-intuitive.

Two beagles, we know that at least one is male. The possible combinations:

MM
FF
MF
FM

So we know that FF is out leaving us with:

MM
FM
MF

So the chance that the other is a male is 1/3 and that the other is a female is 2/3.

Before you go saying that I'm wrong etc. I have a degree in mathematics and studied statistics all the way through .

 

[aterimperator]

This question is just a tricky way of asking what's the ratio between MM and MF/FM, but if you don't believe me, here's some c++ code to simulate it yourself:



#include
using namespace std;

#include // prototypes for rand and srand
using std::rand;
using std::srand;

#include // time for seeding rand
using std::time;


inline int coinflip()
{
return rand()%2; //returns either 0 or 1 (heads or tails)
}

int main(int argc, char *argv[])
{
int n=1000000; //sample size for simulation
int coinFlipValue[2];
int results[2];
results[0]=0;
results[1]=0;
srand( static_cast( time( 0 ) ) );
for(int i=0; i<n; i++)
{
coinFlipValue[0]=0; // so past values don't affect current results
coinFlipValue[1]=0; // we simply clear the values
coinFlipValue[coinflip()]++; //flip coin 1
coinFlipValue[coinflip()]++; //flip coin 2
// cout << coinFlipValue[0] << endl << coinFlipValue[1] << endl;
// cout << coinflip() << endl;
if(coinFlipValue[0]) //conditional that says "if at least 1 of them is heads"
{
if(1<coinFlipValue[0]) //conditional that says "if both are heads"
{
results[0]++;
}
else //if they're not both heads, than up the tails count instead
{
results[1]++;
}
}
}
cout << "# of heads: " << results[0] << endl;
cout << "# of tails: " << results[1] << endl;
system("PAUSE");
return EXIT_SUCCESS;
}

 

[Samirat]

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

Oh, just to be clear, it is critical that the 45-degree filter be placed between the vertical and horizontal filters, else it'll all just remain opaque.

So, is this the idea? After the light passes through the horizontal filter, the light's polarization is known as horizontal. Therefore none will pass through the vertical filter. But if there is a 45 degree filter after the horizontal filter, the light that passes through will be of unknown polarization, even though it was previously all known to be horizontal, since light passing through a 45 degree filter could be either horizontal or vertical. Therefore, some of the light will pass through the vertical filter. 50 percent of light through the horizontal filter, 25 percent through the 45 degree filter, and 12.5 percent through the final, vertical filter, I'd guess.

Yeah, the 45 degree filter would have to be between, since it essentially scrambles the polarizations by destroying what knowledge you had of them. Not logical, of course, but that's quantum mechanics.

 

[geizr]

Are you sure this is true? Like, if you had a vertical and a horizontal filter, which theoretically prevented all light from passing, it would be opaque, correct? But if you put a 45 degree filter on it, are you saying it would actually become somewhat transparent? Because I'd have to try that. I'm not going to a 3-D movie, though, I'll find some other way.

Oh, just to be clear, it is critical that the 45-degree filter be placed between the vertical and horizontal filters, else it'll all just remain opaque.

So, is this the idea? After the light passes through the horizontal filter, the light's polarization is known as horizontal. Therefore none will pass through the vertical filter. But if there is a 45 degree filter after the horizontal filter, the light that passes through will be of unknown polarization, even though it was previously all known to be horizontal, since light passing through a 45 degree filter could be either horizontal or vertical. Therefore, some of the light will pass through the vertical filter. 50 percent of light through the horizontal filter, 25 percent through the 45 degree filter, and 12.5 percent through the final, vertical filter, I'd guess.

Yeah, the 45 degree filter would have to be between, since it essentially scrambles the polarizations by destroying what knowledge you had of them. Not logical, of course, but that's quantum mechanics.

You have the basic idea with just one misstep in understanding. The 45-degree filter doesn't make the light have unknown polarization; it makes the light have a 45-degree polarization. However, the 45-degree polarization is split equally between vertical and horizontal polarizations. So, in that sense it does destroy information about the polarization coming out of the horizontal filter, just as the last vertical filter destroys the information about the 45-degree filter.

Like I said, I've always found this a fun trick to demonstrate to people whenever I go to a 3D movie that uses the polarized lenses(which most good ones do), instead of the red/blue lenses(it doesn't work with those cause they are not polarized).

 

[geizr]

In this case with the polarizations, we are using the wave nature of the light. Explaining the truth of the wave-particle duality of nature is...hard. Really, really, freaking hard. And even harder to make real sense of it. You begin to understand it in a back-of-the-mind intuition, but it's impossible to really convey the true reality of it to anyone who is not well experienced with the math and machinery of quantum mechanics(I tried explaining eigenstates and eigenvalues to a friend of mine once. Got about half-way before he asked me to just stop). It seems easy at first to just say that it means things act like a particle and a wave. But, it's not until you start really truly thinking about it that you discover you don't really know what that truly means. We just know how to calculate it.

Its easier to think about it in an abstract way (i.e. via linear algebra [eigenstates/eigenfunctions]) than it actually is to conceptualise it.

That's one of the main problems with the higher level physics. You can calculate the stuff, but it's hella hard to understand what it really means. It takes many years of experience to gain that insight, and there are aspects that you may never gain that insight with(such as wave-particle duality).