Exploring Probabilities: Are Die Really Biased?

[Bubonic Plague]

Recently, i had learned about probabililties. My E.maths teacher had said that to avoid problems, the questions will always use phrases like "unbiased die", etc. But the problem is, aren't all die inherently biased? If i were to use a die which had pits to represent the numbers. Won't the 1 side be heavier then the 6 side?
Even if i were to use a die which used painted dots to represent the numbers, won't the 6 side be heavier then the 1 side?

So can i conclude that because of that, if i were to play a game which involved predicting the number which will appear on the die after every roll, and a pitted die was used, I should bet on the number 6 because the 1 side is heavier, so the number 1 will end up being the bottom more often, causing the number 6 to show up more often. Or do the laws of physics work differently? Or maybe something else?

Just an extra question too. I know that 2^2 is really(couldn't find a sub-script button) = 2 x 2. I also know that 2^1/2 = [squ] 2. But what does 2^1/2 truly mean? I can't seem to visualize it.

 

[jcsd]

A decent dice will be made so they're are no problems with the dots making one side weigh more than the other, that said it's impossible to make a dice without some sort of bias.

 

[jcsd]

x² = x*x

x³= x*x*x

etc.

x = x^1/2*x^1/2

x = x^1/3*x^1/3*x^1/3

etc.

x⁰ = x/x

x¹ = x

 

[FZ+]

And in that language, x^pi would be?

 

[jcsd]

Originally posted by FZ+
And in that language, x^pi would be?

It'd be ****ing x^pi

 

[Tail]

I recently had the same thoughts about dice. Conclusion: even if there are no dots, a real die will never be completely "unbiased"...

 

[jcsd]

Originally posted by Tail
I recently had the same thoughts about dice. Conclusion: even if there are no dots, a real die will never be completely "unbiased"...

But when you're doing statistics you might as well consider dice to be unbiased as for a decent div=ce you'd only be wrong 1/1000th of the time.

 

[Tail]

Yes, you can. Especially as the force of a die being thrown will make the small differences between the weight of the sides virtually unimportant...

 

[Bubonic Plague]

x2 = x*x

x3= x*x*x

etc.

x = x1/2*x1/2

x = x1/3*x1/3*x1/3

etc.

x0 = x/x

x1 = x

I'm familiar with indices. But you still aren't explaining what x^1/2 really means.

Also, why is it that some numbers in decimals cannot be represented in fraction form?

 

[jcsd]

Okay Bubonic Plague, I'm sure you've heard of the rule: (xⁿ)^m= x^nm, for example (x²)³ = x⁶, using this rule we can see that (x^1/2)² = x.

basically x^1/2 is the inverse power of x², so that if you perform one operation it will be reversed by performing the other.

 

[plus]

Originally posted by jcsd
But when you're doing statistics you might as well consider dice to be unbiased as for a decent div=ce you'd only be wrong 1/1000th of the time.

In terms of the point about paint, you could take some plastic out of the sides to make it weigh the same, and have the paint the same density of the paint/plastic.

However, you could never make a totally unbiased die, as it would be impossible to exactly place each atom due to uncertainty principle (as we can easily measure the momentum to be less than a certain value).

 

[jcsd]

Originally posted by plus
In terms of the point about paint, you could take some plastic out of the sides to make it weigh the same, and have the paint the same density of the paint/plastic.

However, you could never make a totally unbiased die, as it would be impossible to exactly place each atom due to uncertainty principle (as we can easily measure the momentum to be less than a certain value).

casino dice maunfacturers take out little bits of the dice where they put the paint in and they use paint of the same density as the dice.

 

[Bubonic Plague]

Okay Bubonic Plague, I'm sure you've heard of the rule: (xn)m= xnm, for example (x2)3 = x6, using this rule we can see that (x1/2)2 = x.

Yes, i understand this. Maybe i am not bringing my question across properly.

2^2 = 2*2 right?
So how can 2^1/2 be expressed in this form? (This is where the problem lies, i can't visualize 2^1/2 in this form.)

2^2 = 2*2
2^1/2 = ?

Am i being too thick?

casino dice maunfacturers take out little bits of the dice where they put the paint in and they use paint of the same density as the dice.

I never thought of that. So die can be "unbiased".

 

[Hurkyl]

(for x >= 0) x^(1/2) really means the nonnegative quantity such that ( x^(1/2) )^2 = x

aka

x^(1/2) * x^(1/2) = x

 

[Bubonic Plague]

(for x >= 0) x^(1/2) really means the nonnegative quantity such that ( x^(1/2) )^2 = x

So what you are trying to say is that i should see 2^1/2 as [squ]2 rather then 2^1/2 in order to visualize it?

 

[jcsd]

Originally posted by Bubonic Plague
Yes, i understand this. Maybe i am not bringing my question across properly.

2^2 = 2*2 right?
So how can 2^1/2 be expressed in this form? (This is where the problem lies, i can't visualize 2^1/2 in this form.)

2^2 = 2*2
2^1/2 = ?



I never thought of that. So die can be "unbiased".

Well as it's the inverse of the function:

2² = 2*2

but

2 = 2^1/2*2^1/2


The methods for finding the square root of a given number are either iterative or involve infinite series here's a list of them:

http://www.rism.com/Trig/square.htm


Even a casino dice is not truly unbiased, if you tested it enough times you would be able to see and quantify a bias.

 

[Hurkyl]

So what you are trying to say is that i should see 2^1/2 as [squ]2 rather then 2^1/2 in order to visualize it?

I'm not 100% sure what you mean by this, since [squ]2 and 2^(1/2) are the same thing...

If you mean that you shouldn't be thinking of 2^(1/2) as 2 multiplied by itself a bunch of times, then that's probably correct.


The way, IMHO, you should be thinking is just to think of ^ as yet another basic operation, just like you think of * as its own entity instead of mentally thinking of adding up the same number a bunch of times, and you think of + as its own entity instead of mentally thinking of incrementing a number a bunch of times.

 

[FZ+]

Seriously though Hurkyl, how can you talk about something like x^pi in such terms? What does x^pi really represent? Or does it actually represent anything other that x^pi?

 

[Hurkyl]

If you really want a constructive definition of x^π, you could use:

x^y→x^π as y→π (y rational)

That is, choose your favorite constructive definition of x^y with y a rational number, then x^π is the number approximated when y is an approximation of π

 

[jcsd]

If there's a way of construing x^Ï€ at a more intutive level I've not found it, unless you look at it as a point of the curve of some function n^x.

Of course pi does come up as a power for some identities for example off the top of my head iⁱ = e^-Ï€/2