Print

Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

[Witherweaver]

I find the arrogance some here are showing in this argument annoying. WW is trying to understand what you mean, you're not helping him at all. I still don't fully get it. I think vertices should matter, for example: In an ideal die throw, we can assume that the die is randomly rotated, with a random momentum, when it hits the table. The by far most likely case is that the die hits the table on a vertex. Not all vertices are "the same", so it is not clear why we would get a uniform distribution going from here.

This is my thought.  Though in the ten-sided die case, you have two (three?) sets of equivalent vertices.

But, really, I think generally "fair" is simply defined to be symmetric with respect to its faces.

[Mic Qsenoch]

I find the arrogance some here are showing in this argument annoying. WW is trying to understand what you mean, you're not helping him at all. I still don't fully get it. I think vertices should matter, for example: In an ideal die throw, we can assume that the die is randomly rotated, with a random momentum, when it hits the table. The by far most likely case is that the die hits the table on a vertex. Not all vertices are "the same", so it is not clear why we would get a uniform distribution going from here.

But each face relates to the same number/type of vertices in exactly the same way, as already pointed out the faces are indistinguishable.

If you make the assumption that the physical process of rolling erases any biases in the initial state then the distribution is obviously uniform. If you don't make this assumption, no dice has a uniform distribution, not even the regular ones.

[singletee]

Suppose an idealized D10 does not produce a uniform distribution. In that case, there is one face that is at least as likely as any other, and more likely than at least one other. Without loss of generality we can assume that 1 is more likely to be rolled than 2. (Our labeling is arbitrary, not necessarily the standard D10 labeling.) Now we rotate this die such that the 2 face is now where the 1 face started out. Since the physical structure of the die is the same, the probability of rolling any given face is only dependent on its position on the die. So the 2 face ends up in the 1 spot, and the 1 face ends up somewhere other than the 1 spot. Thus 2 is at least as likely as 1, and we have a contradiction. Therefore an idealized D10 produces a uniform distribution.

[qmech]

It's certainly possible to have a die with two faces that are the same shape, but with different chances of landing on each.  However, a d10 has much more symmetry than this: the faces are indistinguishable. 

Suppose it's our job to generate random numbers from 1 to 10.  I pick a d10 out of the box, then put it on the table in front of you.  You then paint it in the standard pattern, starting by putting a 1 on the face on the top.  I then roll the d10 and report the result.

Your job is harder than mine, so I tend to get bored easily.  To make things more interesting I start rolling the blank d10s to decide which way up to give them to you, rather than just taking them our of the box and putting them onto the table.  After you've painted the numbers on I roll the d10 and report the result as before.

In one sense this process is different from the original one: each face of each d10 is now getting a random number, with all possibilities equally likely, rather than the one it was destined to have when it was sitting in the box.  But it doesn't really matter; since blank d10's all look the same whichever way up you put them, you won't even know I've been rolling the d10s before giving them to you if I do it in the next room.  So it can't be the case that my pre-rolling has had any effect on the numbers I end up reporting.  But whichever face was destined to come up at the final roll, it now has a random number on it rather than the one it was supposed to: so our process must have been producing random numbers all along.

[Kuildeous]

Why do you think a d10's results are uniformly distributed?  It isn't regular.

Like I said, you have two equal chances of landing on the odd or even side. You then have five equal chances of getting 1, 3, 5, 7, 9 or 2, 4, 6, 8, 0, depending on what the first one was. Of course, this assumes a perfectly cut d10 so that all the planes are congruent with congruent angles between them, but we have to make that assumption for all dice anyway.

I'm not sure how I can add to that. Without trying to sound like a cop-out, why would not being regular preclude even distribution? How does a d10 not achieve even distribution?

It's the "equal chance" part that isn't clear.  You're just restating 'uniform'.

That's what "uniform" means. An equal chance of each result.

Yes I know this.  My point is Kuildeous is not explaining, he's restating.

Which is why I said that I wasn't sure how to add to that. All I could do was restate it with a little more clarification. I figured I failed to convey an aha moment, and I attempted lamely to find that moment. Obviously I didn't, and I'm not surprised. That's why I had to shift the burden to you. The concept seemed so obvious in my mind that the opposite was impossible for me to fathom. So I asked for your viewpoint so I could possibly understand it.

And I'm not treating you as an idiot. Or at least I hope I didn't look like I was. I've been in this position before with the Monty Hall puzzle. I was as convinced as I am about this d10 matter that the odds of winning increases to 1/2 that I could not fathom that the real odds were actually 2/3. At least not until I had someone explain his viewpoint so I could understand it (and see how wrong I was).

And yes, how you roll the die can affect the randomness of it. If you hold the die with the 4 face facing upward and flick your wrist the same way each time against a surface that is 84 degrees to the table with green felt, then you probably will not get a uniform distribution. But you would have that problem with regular solids too. I take random die rolls to mean that you haphazardly toss the die in a fashion that is not similar to the previous rolls. You hold it differently, you throw with a different angle, you throw with a different speed, you roll on a different surface, you bounce it off of an object, and so on. I suppose this is getting into entropy? Unless you're specifically trying to influence how the die lands, it should have enough chaos in the throw that you can treat it as true randomness.

So barring weirdness in how vertices are struck, as long as there is sufficient distance and spin applied to the die, is there a reason why the distribution of a d10 should not be uniform vs. any of the platonic solids?

[Witherweaver]

....

I understood what you were saying, but you started out with "these have an equal chance of occurring," which is exactly what was under question.  Anyway, it wasn't your posts to which I was referring.

To me the question is not why should it not be uniform, but rather why should it be uniform.   

I think what Mic Q, singletee, and gmech said cover it. 

This is interesting, by the way:

http://mathoverflow.net/questions/46684/fair-but-irregular-polyhedral-dice

Also:

http://www-stat.stanford.edu/~cgates/PERSI/papers/fairdice.pdf

[pacovf]

But, please, keep treating me like an idiot, because that's fun.

I am sorry if I came out that way. I didn't understand that you were talking about the influence of the initial configuration on the final configuration. But I have a feeling that you have to assume an initial configuration that is itself uniform, otherwise you would be cheating. Then your problem disappears?

[Witherweaver]

But, please, keep treating me like an idiot, because that's fun.

I am sorry if I came out that way. I didn't understand that you were talking about the influence of the initial configuration on the final configuration. But I have a feeling that you have to assume an initial configuration that is itself uniform, otherwise you would be cheating. Then your problem disappears?

I think the essence of what I needed  (to make it 'click' for me) was this:

But each face relates to the same number/type of vertices in exactly the same way, as already pointed out the faces are indistinguishable.

I mean my general thought was: not all the vertices and edges are the same, what if they affect the actual roll in nonuniform ways?

[LastFootnote]

I was treating you like a troll, not an idiot. But I apologize for that, too.

I guess in my mind, there is absolutely no reason the burden of proof should be on anybody but you in this case. If you don't think a d10 has a uniform probability distribution, prove why that is.

[Witherweaver]

I was treating you like a troll, not an idiot. But I apologize for that, too.

I guess in my mind, there is absolutely no reason the burden of proof should be on anybody but you in this case. If you don't think a d10 has a uniform probability distribution, prove why that is.

Well I disagree there.  I mean, it's obviously not the same as the other dice:  4, 6, 8, 10, 12, 20.  One of these is different than the others.  Turns out the difference isn't significant here, but I don't think it's dismissively obvious. 

[Witherweaver]

And also when I was posting earlier this morning I was doing other stuff simultaneously, so couldn't really explain things well, so sorry.

[liopoil]

Burden of proof was definitely not on witherweaver. He wasn't arguing that it wasn't uniform, just that it had yet to be shown that it was uniform.

[Witherweaver]

Burden of proof was definitely not on witherweaver. He wasn't arguing that it wasn't uniform, just that it had yet to be shown that it was uniform.

Right, yes.  I may not have been typing clearly though.

[LastFootnote]

Then could somebody please prove to me that a regular polyhedron does have a uniform distribution? Witherweaver does appear to take that as a given.

[liopoil]

Then could somebody please prove to me that a regular polyhedron does have a uniform distribution? Witherweaver does appear to take that as a given.

Singletee's argument works for any die with indistinguishable faces, including all regular polyhedra:

Suppose an idealized D10 does not produce a uniform distribution. In that case, there is one face that is at least as likely as any other, and more likely than at least one other. Without loss of generality we can assume that 1 is more likely to be rolled than 2. (Our labeling is arbitrary, not necessarily the standard D10 labeling.) Now we rotate this die such that the 2 face is now where the 1 face started out. Since the physical structure of the die is the same, the probability of rolling any given face is only dependent on its position on the die. So the 2 face ends up in the 1 spot, and the 1 face ends up somewhere other than the 1 spot. Thus 2 is at least as likely as 1, and we have a contradiction. Therefore an idealized D10 produces a uniform distribution.

[Tables]

Today I visited the Coventry Transport Museum with my sister. It was fun. I also decided to be a total rebel.

[sudgy]

I'm just going to put this here, because me and my brother keep trolling each other with it:

http://imgur.com/OnlBGBh

[Awaclus]

Based on the results of this poll, I'd say that natural stupidity is a far greater threat to humanity.

(the poll was originally on The Telegraph)

[liopoil]

I'm not sure what you mean. The answer is obviously yes, right?

[Witherweaver]

Do we even know that a human made that poll?  Or that only humans voted in it?!  AI already outnumbers us!

[ashersky]

I'm not sure what you mean. The answer is obviously yes, right?

This is correct.  Regardless of the possibility, humans should be worried about all possible threats in the universe.  Aliens, zombies, vampires, robots, pirates.  Be worried an appropriate amount and be done.

[Roadrunner7671]

Do we even know that a human made that poll?  Or that only humans voted in it?!  AI already outnumbers us!

I firmly believe that WW made this post so that we won't expect him to be AI, but I'm onto him.

Vote: WW

[Axxle]

I'm not sure what you mean. The answer is obviously yes, right?

This is correct.  Regardless of the possibility, humans should be worried about all possible threats in the universe.  Aliens, zombies, vampires, robots, pirates.  Be worried an appropriate amount and be done.

This is venturing into RSP territory, but AI is getting to the point where they are replacing jobs and not really opening up too many more. That's what's worrying. We'll have to rethink how the world works if people can't realistically be expected to be employed.

[Kirian]

I'm not sure what you mean. The answer is obviously yes, right?

This is correct.  Regardless of the possibility, humans should be worried about all possible threats in the universe.  Aliens, zombies, vampires, robots, pirates.  Be worried an appropriate amount and be done.

This is venturing into RSP territory, but AI is getting to the point where they are replacing jobs and not really opening up too many more. That's what's worrying. We'll have to rethink how the world works if people can't realistically be expected to be employed.

Relevant:

[gkrieg13]

I'm not sure what you mean. The answer is obviously yes, right?

This is correct.  Regardless of the possibility, humans should be worried about all possible threats in the universe.  Aliens, zombies, vampires, robots, pirates.  Be worried an appropriate amount and be done.

This is venturing into RSP territory, but AI is getting to the point where they are replacing jobs and not really opening up too many more. That's what's worrying. We'll have to rethink how the world works if people can't realistically be expected to be employed.

Or you can program the AI.  Then you'll have lots of jobs!