Probability and infinity

[C1ay]

You wanna try a shot at answering Agen's quesion?

 

He already did. In terms of limits, δ is the infinitesimal probability Agen is seeking...

[arkain101]

Let's say that there is an infinity amount of objects and only one of those objects is the "right" one. What would be the chances of the "right" one being picked?

 

Infinity is mental concept. It is impossible to have an infinity amount of objects. Why? Because infinity is equivelent to zero.. In what way? In the form of 100% equality.

 

As such infinity is a form of oneness, all the same and equall.

 

For example. A universe with infinity atoms would be nothing but ATOMS. The relationship to attain infinity atoms would require absolute zero space (macroscopically).

 

I visualise this like so. A universe of infinity is simple to imagine a universe of neverending bright white light. In this case something is there, but it is undefineable, unmeasureable, and all that exists.

 

As for 'nothing'.. it can be imagined like a universe of never ending black. In this case nothig is there..including an observer.

 

Although there is more to explaining this.. I would make the conclusion that the universe is not infinite..

 

However, something is, and that is the reference frame of light (infinite as in a perspect of one and all from beginning to end), and when something has an infinite quality it can be expected to have very strange and unique behavior. Firstly, it is akin to nothingness.. you can't grab it. It has no boundry or restriction on behavior - it can be particle like or wave like.

[UncleAl]

Which infinity? The number of integers is countable - each one is numbered. The number of points on a line is infinitely larger and uncountable. The number of functions through a point is infinitely larger still.

 

Draw a line segment. Touch it with a(n infinitely sharp pencil) point. If the chance of touching a single point within an infinitely infinite number of other points is indeed zero - or even small - how did you accomplish the task? Do it again, and again, and again... Do you ever miss hitting a point on the line?

 

How many times have you won the lottery?

[Pyrotex]

Infinity is mental concept. It is impossible to have an infinity amount of objects. Why? Because infinity is equivelent to zero.. In what way? In the form of 100% equality.

As such infinity is a form of oneness, all the same and equall.

For example. A universe with infinity atoms would be nothing but ATOMS. The relationship to attain infinity atoms would require absolute zero space (macroscopically).

I visualise this like so. ....

I'm sorry Arkain, but this all makes very little sense. I thought you were on the right track up through "an infinity of objects". But when you said "infinity is equivalent to zero" you left the tracks, wandered through the Twilight Zone, got lost in Never-Never Land, and then contradicted yourself with "universe with infinity atoms".

 

If you were trying to be Mystical, I want you to know you went over the top.

Waaaaaaaaaaaay over.

[LaurieAG]

excellent! good point. I should have remembered that. You wanna try a shot at answering Agen's quesion?

 

Thanks Pyro,

 

I'll just try to put it into a context where its relevance/irrelevance can be put in a slightly different light.

 

Say you had the equation f(x)=y=1 i.e. x=y=1

 

Considering that this equation describes the point 1,1 what would you get if you derived the limits of f(x) from - infinity to 1-delta and the limits from 1+delta to infinity?

 

While you might think that the answer is 2*(infinity-delta) and if you divide by 2 you get your scenario of infinity-delta the function is only for the point x=y=1 not the complete continuum of -infinity to +infinity.

 

In this case the answer is really zero because 1+/- delta is not part of the function and the limits are spurious, mainly because you are talking about one real solution amongst other infinite possibilities that are wrong and therefore not valid solutions to the function. Probability and infinity don't come into it.

[Pyrotex]

:( hunh? :eek:

 

I'll get back to you on this...

[LaurieAG]

:tongue: hunh?

I'll get back to you on this...

 

Hi Pyro,

 

This is probably the bit that's confusing.

 

While you might think that the answer is 2*(infinity-delta) and if you divide by 2 you get your scenario of infinity-delta the function is only for the point x=y=1 not the complete continuum of -infinity to +infinity.

 

You might think this is the answer if the function were f(x)=y=x (not x=y=1 as I stated originally) i.e. one's a point and the other's a line. It's just another variation of the same theme of probability and infinity not being applicable if there is only 1 real solution (all the others are not probable therefore probability is a real misnomer).

[Qfwfq]

Formally, the question Agen poses is the matter of normalization of a probability distribution. If the distribution is complete (covers the whole set of possible outcomes), so it is certain that the outcome must be one of these, which are mutually exclusive, the sum of probabilities must equal 1.

 

I see it as a mathematical matter, not a physical one, so I won't discuss things such as the number of atoms in the universe. If we consider it as a distribution over [imath]\mathbb{N}[/imath], the normalization problem has no non-zero solution unless the distribution is a convergent series (including ones which are non-zero only over a finite subset). This is what the paradox is due to.

Sort out the negations there. :hihi:

 

In the specific case of a flat distribution [math]a_i=a,\;\forall i:\;\sum_ia_i=1\Rightarrow a=0[/math], and yet [math]\sum_i0=0\neq1[/math]

[Pyrotex]

ahhh... ummm... you guys go on ahead. I... uh... need to tie my shoelaces.

[Qfwfq]

Actually, I tied my own laces very sloppy indeed. :doh:

 

I should have said that, in the specific case of a flat distribution [imath]a_i=a,\;\forall i\in\mathbb{N}[/imath], no finite value of [imath]a\neq 0[/imath] will give a finite value of

 

[math]\sum_ia_i[/math]

 

and since [imath]\sum_i0=0[/imath], strictly there is no solution for normalization of the distribution. What I posted yesterday is, however lax, often taken as an intuitive solution and is certainly the limit for a sequence of flat distributions with an increasing number of outcomes. It is therefore true "in a sense" despite the fact that the distribution of zero probabilities can't sum up to unity.

 

In short, it is a singularity in the mathematical framework.