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CodeSlinger
2007-Sep-10, 03:11 AM
Suppose you had a random number generator that was capable of generating real numbers of infinite precision (this would be something better than current computer PRNG's, which can only produce numbers of fixed precision, and aren't really truly random). What is the probability of this RNG generating an integer-like number (something like 24.0)? The number of integers and real numbers that can be generated are both infinite, but intuitively, the infinity of integers should be "smaller" than the infinity of real numbers.

Jens
2007-Sep-10, 03:26 AM
My intuition is that there there would be no possibility, because there are infinitely more real numbers than integers. Of course, no computer could ever do that, so the question may be meaningless.

01101001
2007-Sep-10, 03:48 AM
My intuition is that there there would be no possibility, because there are infinitely more real numbers than integers.

Yeah. That would be the same probability of its generating a number with a fractional part of .999...

Delvo
2007-Sep-10, 04:24 AM
But then, each time you use the thing, whatever number you got, the odds of getting that number were zero before it happened.

CodeSlinger
2007-Sep-10, 05:37 AM
Can the probability of getting any number be zero? With a hypothetical perfect RNG, the probability of getting any number should be equal to the probability of getting any other number. So if the probability of getting one number is zero, then they would all be zero. Doesn't this mean the RNG would spit out NO numbers whatsoever? But it does (or should), so the probability should be a small but non-zero number.

Jens, yes, I do agree that no computer may ever be able to serve as such as perfect RNG. But for the purposes of this exercise, let's pretend that we do have such a thing. Let's say it's the dice that God plays with :)

Ken G
2007-Sep-10, 05:53 AM
You could easily make this a precise question simply by increasing both the precision and range and watching what happens to the probability of getting an integer. The probability is simply the range over the precision, it makes no difference how large each of those is. Thus you can't just say "both are infinite", because what will always matter is their ratio.

mfumbesi
2007-Sep-10, 06:46 AM
If you have a "true" random number generator and your range is infinity, I think the probabilities of generating a number say 10.0 (Or any number) is zero Or alternatively 1 to infinite, which realistically is zero.

Ken G
2007-Sep-10, 06:51 AM
Any specific number, yes, but as for getting an integer, it's the ratio of range to precision. If I have a precision that only includes one place after the decimal point, my range can be anything you like, even infinite (in principle, of course that's impossible in practice), and the chance of getting an integer is always 1/10. If I have two decimal places, it's 1/100, etc. But my full precision must include all the places to the left of the decimal as well, so that's where the range comes in. Thus getting back to the OP, even if I have infinite precision and infinite range, I still have to specify how much of that infinite precision is used on the left of the decimal place, and how much on the right, in terms of a ratio.

astromark
2007-Sep-10, 06:57 AM
quote, "The number of integers and real numbers that can be generated are both infinite, but intuitively, the infinity of integers should be "smaller" than the infinity of real numbers. end quote"... Umm....

Hay wait a minite... that can't be right, can it?
Infinite is infinite... There can be no 'larger' portion of infinite possibilities. Any less than infinite, is not infinite. How did you let this go?

Ken G
2007-Sep-10, 07:11 AM
There are indeed different "levels of infinity". For example, there are an infinite number of rational numbers between 0 and 1, and an infinite number of irrationals there as well, but if you truly pick a random point on a line from 0 to 1, it will always be an irrational number. This is because the irrationals are 'uncountably infinite' and the rationals are 'countably infinite'. The latter means that since rationals can be expressed as fractions, you can count them (indeed, overcount them) by going 1/1, 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5,... see the pattern? But you cannot do that with the irrationals. Or another way to think of that, rationals have repeating decimals, irrationals don't -- so the latter are arbitrarily more numerous in a given range with arbitrary precision. But the OP also specified that the range was arbitrarily large, so that's why we haven't enough information to tell how likely the integers, or the rationals, will be compared to the irrationals.

Kullat Nunu
2007-Sep-10, 11:06 AM
What is the probability of this RNG generating an integer-like number (something like 24.0)?

Infinitely small, that is 0. The probability of a given value in a continuous probability function is always 0, even if the function is limited.


The number of integers and real numbers that can be generated are both infinite, but intuitively, the infinity of integers should be "smaller" than the infinity of real numbers.

Your intuition is right, there are "different-sized" infinities. The size of an infinity is called cardinality. The cardinality of the set of real numbers, Aleph-1, is larger than the cardinality of the set of integers, Aleph-0 which is the smallest possible infinity. Interestingly, one can't prove mathematically that there are intermediate cardinalities between those two.

JohnD
2007-Sep-10, 01:02 PM
You're getting hung up on infinity.
Whatever the odds are for getting, say, 23.99999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999 etc.etc., they are exactly the same as getting 24. Of course that first number is not infinitely precise, but anyway, I hope the point is made.

This fallacy prevents peole using "1 2 3 4 5 6" as their lottery number, when the odds of that coming up are exactly the same as any other number.

John

a1call
2007-Sep-10, 01:27 PM
Probability of getting anything in an infinite set is:
1/infinity
Which is undefined, not 0. It is the same as divide by 0.
It is a common misunderstanding (http://mathforum.org/library/drmath/view/62486.html).

CodeSlinger
2007-Sep-10, 02:57 PM
Thanks for the responses ladies and gents :) After reading through your replies and a1call's linked page, I *think* I got it. The probability of getting an integer (or any other number) from the hypothetical perfect RNG is simply 1/infinity, an infinitely small number tending to, but *not* equal to, zero. The way I "visualize" this is that as I run (execute, crank through) the RNG more and more times, the ratio of integers versus real numbers will continue to decrease and converge on zero. The same way that as I flip a coin more and more times, the ratio of heads to tails will converge to 1:1.

Ken G, in belated response to your question, I was thinking that the hypothetical perfect RNG would have infinite precision on both the left hand side and the right hand side of the decimal point. As you said, if there is one decimal point precision, the odds would be 1/10, if there are two, the odds would be 1/100. So given infinite precision on the RHS, the odds should be (if I'm thinking about this correctly) 1/infinity. Same result as above, slightly different path. Which I take to be a good sign; if the two reasoning gave different results, my noodles would really be baked!

Thank you, everyone, for humoring me :)

Ken G
2007-Sep-10, 04:53 PM
So given infinite precision on the RHS, the odds should be (if I'm thinking about this correctly) 1/infinity.

That's true if you have finite precision on the LHS-- but that's not the case if you have an infinite range of numbers. You have to specify the digits of precision on both sides, or at least their ratio if they tend to infinity, before you can know what the probability of getting an integer is. There's no escaping the fact that there's no way to actualize an infinite decimal expansion, you need a finite algorithm to generate it and only that algorithm can answer the OP. You are right that the probability of getting any particular number tends to zero as either the range or the precision tends to infinity, for any random algorithm.

CodeSlinger
2007-Sep-10, 04:55 PM
Ok, this part I still don't understand. Why does the precision on the LHS matter in the probability of getting an integer? Only the RHS differentiates integers from non-integers.

Disinfo Agent
2007-Sep-10, 05:01 PM
Suppose you had a random number generator that was capable of generating real numbers of infinite precision (this would be something better than current computer PRNG's, which can only produce numbers of fixed precision, and aren't really truly random). What is the probability of this RNG generating an integer-like number (something like 24.0)? The number of integers and real numbers that can be generated are both infinite, but intuitively, the infinity of integers should be "smaller" than the infinity of real numbers.Your problem is lacking information. You say you have a random number generator, but as soon as you abandon finite probability spaces (and the integers are certainly infinite) you must specify which kind of randomness you're talking about.


Can the probability of getting any number be zero?Yes.


With a hypothetical perfect RNG, the probability of getting any number should be equal to the probability of getting any other number. So if the probability of getting one number is zero, then they would all be zero.It depends on what you mean by "perfect" RNG. The probability need not be zero for all real numbers.


Doesn't this mean the RNG would spit out NO numbers whatsoever?No, it doesn't, actually.

peter eldergill
2007-Sep-10, 05:11 PM
I remember a thread I started a while ago about "flawed questions", if anyone remembers.

Like :"Can Jesus miocrowave a burrito so hot that even He can't eat it?"

(Obviosly not intended to start some sort of religious debate, unless you're Homer Simpson or Ned Flanders)

I'm wondering if this is a flawed question?

I think the probability is 0, as you could probably define a density function and try to calculate the area under the curve at a single point.

Wow..this question is a bugger, isn't it?

But is it a flawed question?

Pete

Note: I suppose you could try to calculate the probability of finding a random integer from the real number system and it would still be 0 (or a limit as x goes to infinity), but how about an interval of real numbers?

CodeSlinger
2007-Sep-10, 05:17 PM
Hello Disinfo Agent,

What are the different kinds of randomness? The kind of randomness I was thinking about for the hypothetical RNG is one where given infinite time, all real numbers would be enumerated. Would this be perfect uniform randomness?

You said that the probability of getting a number can be zero, and went onto say that this does not mean the RNG would spit out no numbers whatsoever. Can this be in the case of a perfect uniform RNG (if that is the right term for what I'm thinking about), where the probability of getting any number should be equal to the probability of any other number? Can an event that has zero probability still occur?

CodeSlinger
2007-Sep-10, 05:22 PM
Note: I suppose you could try to calculate the probability of finding a random integer from the real number system and it would still be 0 (or a limit as x goes to infinity), but how about an interval of real numbers?

The first part is precisely what I was trying to ask in the OP. My current thinking is that it is not zero, but an infinitely small number that tends to zero. For the second part, I think the answer should be the same, since for any given interval there are still an infinite number of real numbers contained within.

Disinfo Agent
2007-Sep-10, 05:23 PM
Hello Disinfo Agent,

What are the different kinds of randomness? The kind of randomness I was thinking about for the hypothetical RNG is one where given infinite time, all real numbers would be enumerated.But there are many different ways of doing that. Infinitely many, as a matter of fact. Infinitely many different RNGs that can be constructed, if you will.


Would this be perfect uniform randomness?Uniform randomness is one kind of randomness, but here's the kicker: with uniform randomness, you'll never manage to span all the integers, only a finite subset of them.


You said that the probability of getting a number can be zero, and went onto say that this does not mean the RNG would spit out no numbers whatsoever. Can this be in the case of a perfect uniform RNG (if that is the right term for what I'm thinking about), where the probability of getting any number should be equal to the probability of any other number? Can an event that has zero probability still occur?Yes to both questions. In infinite spaces, and event with zero probability isn't necessarily impossible; just very, very unlikely.

CodeSlinger
2007-Sep-10, 05:31 PM
But there are many different ways of doing that. Infinitely many, as a matter of fact. Infinitely many different RNGs that can be constructed, if you will.

Cool :) I'm not particularly concerned about how this might be achieved. As long as it is not a logical impossibility (which would render this line of questioning moot), that's good enough for now.


Uniform randomness is one kind of randomness, but here's the kicker: with uniform randomness, you'll never manage to span all the integers, only a finite subset of them.

Given infinite time, though, it would, yes? In any case, I think this is also tangential to the question at hand. My mind is feeble, one brain-twister at a time, please :)


Yes to both questions. In infinite spaces, and event with zero probability isn't necessarily impossible; just very, very unlikely.

How can this be? I thought the very definition of "impossible" is "an event with zero probability". This is what I would like to focus on, if you don't mind.

Disinfo Agent
2007-Sep-10, 05:36 PM
Given infinite time, though, it would, yes?No, with a uniform distribution you're always confined to some bounded interval [a, b]. Even for the positive integers, the simplest infinite set you can think of, there is no distribution (no RNG) that is uniform.


How can this be? I thought the very definition of "impossible" is "an event with zero probability".No, "impossible" just means that it never happens. :)
And in finite spaces that's equivalent to saying that the event has zero probability -- but not in infinite spaces. In infinite spaces, "most" events have zero probability.

Ken G
2007-Sep-10, 05:37 PM
But is it a flawed question?
Yes, not enough information is given about the algorithm that would be needed to actually do this.

Note: I suppose you could try to calculate the probability of finding a random integer from the real number system and it would still be 0 (or a limit as x goes to infinity), but how about an interval of real numbers?
The number of integers in any fixed interval is finite, the number of rationals is countably infinite, and the number of reals is uncountably infinite, so that tells you the hierarchy of likelihoods. But to get an actual probability, you still need to know how the number is being selected. Let's say we take a fixed interval, move a random distance along it, and then measure the point where we stop. Now the probabilities are going to depend on how precise our measurement is. But in the limit as the precision increases (the density of "tickmarks" on our ruler), the chance of getting an integer drops toward zero. Still, we always get a rational result-- we cannot measure something irrational, though we can certainly conceptualize an irrational distance. In some ideal sense, we expect all random distances to be irrational, by the hierarchy, but in practice none of them ever are. The mathematicians can argue it out with Zeno, I want an algorithm.

Disinfo Agent
2007-Sep-10, 05:39 PM
Nevertheless, Ken, a sequence of rational numbers can converge to an irrational number.

Ken G
2007-Sep-10, 05:44 PM
Certainly-- but convergence is not an algorithm for generating that number, it is more like the label of the number. Saying my name is "Ken" is not the same as telling me how that name can get pulled from a hat. I need a finite algorithm to know what we are talking about. I suppose one could say that if we have infinitely precise RNGs we are assuming we can have infinite algorithms, but I prefer to think of such things as a limit as finite algorithms get arbitrarily complicated. I don't know what it means to say "pick a random point on a line segment"-- what is a point and how do you tell which one you've picked?

CodeSlinger
2007-Sep-10, 05:47 PM
No, with a uniform distribution you're always confined to some bounded interval [a, b]. Even for the positive integers, the simplest infinite set you can think of, there is no distribution (no RNG) that is uniform.

I don't think this is true; consider the following algorithm:

Have one program/machine count integers up from 0
Have another program/machine count integers down from -1
Record all output from both programs/machines

Given infinite time, this should give you all integers from infinity to negative infinity (uniformly, no less!).


No, "impossible" just means that it never happens. :)
And in finite spaces that's equivalent to saying that the event has zero probability -- but not in infinite spaces. In infinite spaces, "most" events have zero probability.

My brain just asploded. Can you give me some references to read on this? Preferably at a beginner level please.

Edit: Just realized that the "algorithm" can be simplified to *one* program machine that alternates between counting upward in positive integers and downward in negative integers. So it would spit out output like "0, 1, -1, 2, -2, 3, -3 ...". This would get the job done, for half the sticker price :)

CodeSlinger
2007-Sep-10, 05:51 PM
Hi Ken G, I asked you this earlier, but it probably got lost in the shuffle. I still don't understand why the LHS precision would factor in the probability of fetching an integer from a system of real numbers. Because as far as I can tell, only the RHS differentiates integers from non-integers.

Ken G
2007-Sep-10, 06:00 PM
Hi Ken G, I asked you this earlier, but it probably got lost in the shuffle. I still don't understand why the LHS precision would factor in the probability of fetching an integer from a system of real numbers. Because as far as I can tell, only the RHS differentiates integers from non-integers.You're right that only the number of digits on the right of the decimal will matter for the integer probability, but I'm imagining that that number is determined by the difference between the total number of digits you have to the total number that will be on the left of the decimal, as the latter is what controls the range. So if the range is the number of digits on the left, expressed as a power of 10, and the total precision is the total number of digits, as a power of 10, then the ratio of the former to the latter is the integer probability-- which is also the number of digits on the right, as a negative power of 10. So what I mean is, if the range and precision are both infinite, it's still not clear "how many digits are left over" to be put to the right of the decimal. We might, for example, assert that there's always only one decimal digit, even as the range and precision go to infinity. More information is needed.

CodeSlinger
2007-Sep-10, 06:19 PM
Ok, I see. As I said earlier, I'm positing infinite precision on the RHS. Given this, am I correct in thinking that the probability in question should be 1/infinity?

Ken G
2007-Sep-10, 07:30 PM
Ok, I see. As I said earlier, I'm positing infinite precision on the RHS. Given this, am I correct in thinking that the probability in question should be 1/infinity?
Yes, you're right.

CodeSlinger
2007-Sep-10, 08:15 PM
Thanks again, Ken G, for your patience and your help on this.

Ken G
2007-Sep-10, 11:21 PM
No patience required-- interesting issues.

publius
2007-Sep-11, 12:34 AM
quote, "The number of integers and real numbers that can be generated are both infinite, but intuitively, the infinity of integers should be "smaller" than the infinity of real numbers. end quote"... Umm....

Hay wait a minite... that can't be right, can it?
Infinite is infinite... There can be no 'larger' portion of infinite possibilities. Any less than infinite, is not infinite. How did you let this go?

Actually infinity is not infinity. Well, depends on what you mean by infinity, actually. The familiar "lazy eight" of calculus is really a potential infinity of the real line. There is then the notion of actual infinity. Making infinity real, and not just a potential in the sense of no matter how big something is, increasing it yet some more.

Cantor was developed a theory of infinity, the transfinite numbers,

http://en.wikipedia.org/wiki/Aleph_number

And they are denoted by the Hebrew letter Aleph. Aleph-null is the infinity of the integers, (1, 2, 3, ........). This is known as "countable infinity". The infinity of points on the real line, however, is greater than this. Much greater. :) You cannot put the points on the real line in one-to-one correspondence with the integers.

Now, there are bodies of theory about this that I don't begin to understand. The next infinity is "thought to be", Aleph_1 = 2^(Aleph_0) (This says the real numbers is the power set of the set of integers). However, that cannot be proven nor disproven within some set of axioms, yada, yada, without postulating something else that apparently some don't agree with, or aren't sastified with. It's all beyond my little mind, so I'd better not try to ramble on about it. This is what the high priests of Principia Mathematica incantate about.

Anyway, if that is so, then Aleph_1 is the infinity of the real line. Now, you might think the set of points on the plane is greater than that. It isn't. The set of points in the plane can be put into 1:1 correspondence with the points on the real thing. And the same with the points in 3D space, with 4D space. Even Aleph_0-D space!

To get bigger than Aleph_1, you've got to go to an incountably infinite number of dimensions! That is an Aleph_1 dimensional space.

Does that get you to Aleph_2? I don't have a clue. THe point is that the next infinity in the aleph hierarchy is much more mind boggling infinite than the previous one. We get Aleph_1 by raising 2 to the Aleph_0 power. We don't get Aleph_2 by the same thing. We've got to do something "even bigger".

Now, there is a limit to the Alephs. That is Aleph_infinity. That is the Big Boy. Absolute Infinity. Now were are talking something. This sucker is so big that it is inconceivable. It is That Which Cannot Be Named. It is what it is. That's all you can say. Any property that can be conceived of is held by a lower Aleph.

It is impossible to speak of That Which Cannot Be Named without getting very metaphysical, and going off into La-La land. So I'll simply stop now. :lol:

-Richard

publius
2007-Sep-11, 12:45 AM
PS. Transfinite numbers have some strange properties. Remember, when we're talking transfinite numbers, we are making infinity(ies) *real*. A real mathematical entity that can be dealt with and operated on just like finite numbers. That was Cantor's discovery/creation. And it wasn't without controversy.

Aleph_0 is something called lower case omega, 'w', which I'll use.

1+ w = w. That is, if I'm on the integer line, and I start at 1 rather than 0 and count off forever, I get w just as normal. I've just shifted where I started. However,

w + 1 != w

That means that I counted off to infinity and then I ask for one greater than that. :) "Forever and a day", IOW.

IOW, forever from today is the same as forever from tomorrow. However, forever and another day is something else, one more. :lol: But this stuff is rigorous, make no mistake about it.

-Richard

Disinfo Agent
2007-Sep-11, 09:44 AM
I don't think this is true; consider the following algorithm:

Have one program/machine count integers up from 0
Have another program/machine count integers down from -1
Record all output from both programs/machines

Given infinite time, this should give you all integers from infinity to negative infinity (uniformly, no less!). [...]

Edit: Just realized that the "algorithm" can be simplified to *one* program machine that alternates between counting upward in positive integers and downward in negative integers. So it would spit out output like "0, 1, -1, 2, -2, 3, -3 ...". This would get the job done, for half the sticker price :)There's nothing random about that algorithm, though. It's just an enumeration of the integers, one at a time. But it doesn't give you a random integer at each step of the iteration; one can predict in advance which integer your algorithm will produce after n iterations.


My brain just asploded. Can you give me some references to read on this? Preferably at a beginner level please.Any introductory textbook on Probability or Probability and Statistics should cover these topics with greater or smaller depth. Although not exactly a beginner-level book, I suggest Feller's An Introduction to Probability Theory and Its Applications, vol. II.

Disinfo Agent
2007-Sep-11, 10:07 AM
Now, there is a limit to the Alephs. That is Aleph_infinity. That is the Big Boy. Absolute Infinity. Now were are talking something. This sucker is so big that it is inconceivable. It is That Which Cannot Be Named. It is what it is. That's all you can say. Any property that can be conceived of is held by a lower Aleph.

It is impossible to speak of That Which Cannot Be Named without getting very metaphysical, and going off into La-La land. So I'll simply stop now. :lol:Nice overview of infinite cardinalities here. (http://www.mathacademy.com/pr/minitext/infinity/)
And more (explains Absolute Infinity). (http://dbanach.com/infin.htm)

CodeSlinger
2007-Sep-11, 03:28 PM
There's nothing random about that algorithm, though. It's just an enumeration of the integers, one at a time. But it doesn't give you a random integer at each step of the iteration; one can predict in advance which integer your algorithm will produce after n iterations.

I was trying to address what you said here:


No, with a uniform distribution you're always confined to some bounded interval [a, b]. Even for the positive integers, the simplest infinite set you can think of, there is no distribution (no RNG) that is uniform.

I think my algorithm shows that a uniform distribution is not always confined to a bounded interval. True, my algorithm was in no way not random :) But my intuition is that if it is possible to uniformly enumerate an unbounded interval (which my algorithm can do, given infinite time), it should also be possible to uniformly randomly sample from an unbounded interval, given infinite time.


Any introductory textbook on Probability or Probability and Statistics should cover these topics with greater or smaller depth. Although not exactly a beginner-level book, I suggest Feller's An Introduction to Probability Theory and Its Applications, vol. II.

Ok, I'll keep an eye out for those. But being the cheapskate I am (as well as lazy), I was hoping for an online reference or two. I'm just looking for some resources that expound on the idea that events with 0 probability can still occur.

a1call
2007-Sep-11, 04:33 PM
I'm just looking for some resources that expound on the idea that events with 0 probability can still occur.

1st off some posts here are way above my head, but if you are saying what I think you are, then:

Chance of pulling any given integer (say 361) from a set of infinite integers is not 0. It is greater than 0 but next to impossible. Chances are that within any given finite time period no matter how long, you will not pull the number 361.
However you will pull "an" integer which was as unlikely to pull as the number 361.

Hard to grasp perhaps but that's only because infinity is an incomprehensible concept. Everything we know including the number of neurons we use to know them with is finite.

CodeSlinger
2007-Sep-11, 04:56 PM
a1call, I'm totally with you. I too believe that the odds would be infinitely small but non-zero. What I'm asking for references on is the following:


In infinite spaces, an event with zero probability isn't necessarily impossible; just very, very unlikely.

And when I asked, "isn't impossible defined as having zero probability?", he replied:


No, "impossible" just means that it never happens. :)
And in finite spaces that's equivalent to saying that the event has zero probability -- but not in infinite spaces. In infinite spaces, "most" events have zero probability.

...which totally blows my mind, hence the request for references.

01101001
2007-Sep-11, 05:05 PM
Chance of pulling any given integer (say 361) from a set of infinite integers is not 0. It is greater than 0 but next to impossible.

And infinity, being somewhat on the large side, means even the probability of pulling a 3-digit number, or even a number having a googolplex or fewer digits, is on the same order. Whatever finite subset you might consider, is inifinitesimally smaller than the infinite remainder.

a1call
2007-Sep-11, 07:07 PM
This link (http://www.jcu.edu/math/vignettes/infinity.htm) pointed out by Hortonheardawho here (http://www.marsroverblog.com/dyn/entry/52882/discussion_page/41)
and is hosted on server which seems to be overloaded at the moment, might be of interest.

Disinfo Agent
2007-Sep-11, 07:35 PM
I think my algorithm shows that a uniform distribution is not always confined to a bounded interval.No, it doesn't, because your algorithm does not represent a distribution, in the sense of probability theory.


True, my algorithm was in no way not random :)It was in no way random! That's why it's not a distribution.


But my intuition is that if it is possible to uniformly enumerate an unbounded interval (which my algorithm can do, given infinite time), it should also be possible to uniformly randomly sample from an unbounded interval, given infinite time.Your intuition is wrong, sorry. Here's why:

Suppose there was a uniform distribution on the integers. Then each integer would have some positive probability of turning up; let's call that probability c. Now, one of the fundamental properties of probability is that when you add up the probabilities of all possible outcomes, you get 1 (100%), right?

But unfortunately

c + c + c + ... + c + ... (summed for the infinity of integers) = infinity

So there can be no such positive probability c.


But being the cheapskate I am (as well as lazy), I was hoping for an online reference or two. I'm just looking for some resources that expound on the idea that events with 0 probability can still occur.Look for the keywords "event of zero probability", and the like.

BioSci
2007-Sep-11, 08:00 PM
Suppose there was a uniform distribution on the integers. Then each integer would have some positive probability of turning up; let's call that probability c. Now, one of the fundamental properties of probability is that when you add up the probabilities of all possible outcomes, you get 1 (100%), right?

But unfortunately

c + c + c + ... + c + ... (summed for the infinity of integers) = infinity

So there can be no such positive probability c. Since a probability must be non-negative, it can only be c=0.


I think this demonstrates the problem with trying to assign a "probability" or even trying to identify a "random" integer. The problem is that infinty is better thought of as a concept and not a number. In this example "c" can not be a positive number - but it also can not be 0 since according to probability the sum of individual probabilities must sum to 1!

Trying to define 1/infinity is just like trying to define 1/0 - it doesn't work and has no meaning. There is no such thing as a "random" member of an infinite set - how could you ever establish that any such number (or a set of such numbers) was random ? - such a concept can have no meaning when selected from an infinite set. There is no way or method to select such a "random" number without limiting yourself to a finite (and therefore non-random) selection.

The question of the probability of randomly selecting a specific number from an infinite set is ill-formed and has no meaning!

Disinfo Agent
2007-Sep-11, 08:06 PM
I think this demonstrates the problem with trying to assign a "probability" or even trying to identify a "random" integer. The problem is that infinty is better thought of as a concept and not a number. In this example "c" can not be a positive number - but it also can not be 0 since according to probability the sum of individual probabilities must sum to 1!You're right that it cannot be zero in the specific case of the integers. However, it can be zero in other contexts, such as the real numbers.


The question of the probability of randomly selecting a specific number from an infinite set is ill-formed and has no meaning!Nonsense. Look, how about you guys research things a little before pontificating about them?

CodeSlinger
2007-Sep-11, 08:57 PM
No, it doesn't, because your algorithm does not represent a distribution, in the sense of probability theory.

Sorry about that, I was unaware of exactly what constitutes a probability distribution.


It was in no way random! That's why it's not a distribution.

My apologies again. The "not" was a stowaway in my sentence introduced by sloppy editing...


Look for the keywords "event of zero probability", and the like.

Thanks to that tip, I found:

Probability Paradox (http://www.cs.cmu.edu/~ndr/ProbabilityParadox.html)

Which finally managed to get through my thick skull why probability 0 != impossible in an infinite space.


Look, how about you guys research things a little before pontificating about them?

Ouch, that stings... DA, I understand your frustration. As unlikely as it seems, I've also been in the position of trying to explain to people things that they could've easily researched on their own. I *have* tried to research this on my own, but it's difficult without pointers in the right direction. Which is why I ended up posting my questions here. I greatly appreciate your patience thus far, and hope you won't mind more dumb questions.

I still don't understand why there can be no uniform distribution on an unbounded interval. I understood your explanation from the previous post. The problem is, it seems to me that it would apply to, and disqualify, *any* distribution on an unbounded interval. But there are clearly discrete distributions with infinite support (http://en.wikipedia.org/wiki/Probability_distribution#With_infinite_support) and continous distributions with infinite support (http://en.wikipedia.org/wiki/Probability_distribution#Supported_on_the_whole_re al_line). So what is it that makes a uniform distribution (either discrete or continous) with infinite support impossible?

Disinfo Agent
2007-Sep-11, 09:45 PM
Sorry about that, I was unaware of exactly what constitutes a probability distribution.Without getting too technical, it basically means a function which gives you a rule to generate random numbers (as many or as few as you like).

I should clarify now that although we've been using the term RNG informally in this thread, strictly speaking these "rules" are not algorithms. Whenever you have an algorithm, that means you can know in advance exactly which number you will get at iteration n, which is obviously at odds with the notion of "random".

In finite spaces -- that is, when the total number of different possible outcomes of a random experiment is an integer -- it's fairly intuitive that "at random" means that each possible outcome is assigned the same probability. For example, if the experiment consists of throwing a die, each side should have probability 1/6 when the throw is random.

...Well, actually it's not that simple. What if the die has some bias which makes one of the sides (say, the number 4) turn up slightly more often than the others, but still all sides are possible outcomes? This should also be random!

We call the first kind of randomness, where all outcomes are equally likely, a uniform distribution. You can no doubt see that there will be many more, different non-uniform distributions even in this very simple setup.


I still don't understand why there can be no uniform distribution on an unbounded interval. I understood your explanation from the previous post.The reasoning is similar. If a set is unbounded, then it can be written as a union of "smaller" bounded sets. For example, the set R of the real numbers can be written as:

... U [-1, 0[ U [0, 1[ U [1, 2[ U ...

You can probably pick it up from here...


I *have* tried to research this on my own, but it's difficult without pointers in the right direction. Which is why I ended up posting my questions here.Then why not just say so?

Why a number chosen at random from the reals has zero probability. (http://people.hofstra.edu/Stefan_Waner/cprob/cprob2.html) (--> usually)
Same question at Ask DrMath (http://mathforum.org/library/drmath/view/62719.html)
Further reading: "Almost surely" at Wikipedia (http://en.wikipedia.org/wiki/Almost_surely)
"Random number generator" at Wikipedia (http://en.wikipedia.org/wiki/Random_number_generator)

CodeSlinger
2007-Sep-11, 10:29 PM
Thank you for the definition of distributions and the links :)

I think I did a poor job of asking my question in the last part of my previous comment.


Suppose there was a uniform distribution on the integers. Then each integer would have some positive probability of turning up; let's call that probability c. Now, one of the fundamental properties of probability is that when you add up the probabilities of all possible outcomes, you get 1 (100%), right?

But unfortunately

c + c + c + ... + c + ... (summed for the infinity of integers) = infinity

So there can be no such positive probability c.

This I follow, and I understand how this shows one cannot have a uniform distribution on integers. But with a little re-wording, I can apply the same argument to, say, the Boltzmann distribution, and argue that one cannot have a Boltzmann distribution on integers. Except if I'm reading this page correctly, Boltzmann distribution does work on an infinite discrete set such as integers:

http://en.wikipedia.org/wiki/Probability_distribution#With_infinite_support

If the Boltzmann distribution is accepted as one that works with infinite discrete sets, then your argument either does not work or does not apply to it. If it's the case that your argument does not work, then it does not explain why we cannot have a uniform distribution over integers, and I would like to request an alternate explanation of why we cannot have a uniform distribution over integers. If it's the case that your argument does not apply to the Boltzmann distribution, please explain why it does not apply, because I can't see it.

Vadim
2007-Sep-11, 10:41 PM
Any introductory textbook on Probability or Probability and Statistics should cover these topics with greater or smaller depth. Although not exactly a beginner-level book, I suggest Feller's An Introduction to Probability Theory and Its Applications, vol. II.

That is definitely not a beginner's book :) There is also a book by Billingsley, Probability and Measure or something like that, that might be good.

Probably any book will be a bit tough, because these are fairly difficult concepts, especially the first time you deal with them. I didn't see that you ever specified a distribution, which is necessary to answer the question properly. It isn't possible to assign equal probability density to all real numbers. It is possible to do so on a finite interval of real numbers, but if you want to choose from all real numbers, you have to assign different probabilities to different intervals of equal length. But if you choose any continuous probability distribution, the probability of any specific number is zero. Furthermore, the probability of the number chosen being an integer is also zero. The thing that seems to be causing people to say meaningless things like the probability is 1/infinity is the idea that a probability zero event cannot occur. It can. The Feller or Billingsley references talk about things like this.

Vadim
2007-Sep-11, 10:47 PM
This I follow, and I understand how this shows one cannot have a uniform distribution on integers. But with a little re-wording, I can apply the same argument to, say, the Boltzmann distribution, and argue that one cannot have a Boltzmann distribution on integers. Except if I'm reading this page correctly, Boltzmann distribution does work on an infinite discrete set such as integers:

http://en.wikipedia.org/wiki/Probability_distribution#With_infinite_support

If the Boltzmann distribution is accepted as one that works with infinite discrete sets, then your argument either does not work or does not apply to it. If it's the case that your argument does not work, then it does not explain why we cannot have a uniform distribution over integers, and I would like to request an alternate explanation of why we cannot have a uniform distribution over integers. If it's the case that your argument does not apply to the Boltzmann distribution, please explain why it does not apply, because I can't see it.

Disinfo's argument looks good to me. Not familiar with the Boltzmann distribution myself, but if it assigns a probability to each integer (and only to the integers), then those probabilities cannot be equal. That's OK, because an infinite sum can converge to a finite value.

Here is an example - if you repeatedly flip a fair coin, what is the probability distribution of N, where N is the first toss that comes up heads? There is a 0.5 probability N=1, a 0.25 probability N=2, a 0.125 probability N=3, and so on. This assigns probability to every integer; it is possible (very unlikely, but possible) that you will get 1,000,000,000,000,000 tosses all tails before you get a head. But:

0.5+0.25+0.125+0.0625+...(and so on forever)

does not add up to infinity. It adds up to a finite number, which is one. So this is a valid probabililty distribution. You just can't make all the probabilities equal. Then the infinite sum wouldn't converge.

Hope this is helpful. You are asking some subtle questions, and you got some bad answers early on :( Disinfo's answers look good to me though.

Vadim
2007-Sep-11, 10:48 PM
This I follow, and I understand how this shows one cannot have a uniform distribution on integers. But with a little re-wording, I can apply the same argument to, say, the Boltzmann distribution, and argue that one cannot have a Boltzmann distribution on integers. Except if I'm reading this page correctly, Boltzmann distribution does work on an infinite discrete set such as integers:

<linky - I deleted this because I'm new and it looks like I'm not allowed to make a post with links in it>

If the Boltzmann distribution is accepted as one that works with infinite discrete sets, then your argument either does not work or does not apply to it. If it's the case that your argument does not work, then it does not explain why we cannot have a uniform distribution over integers, and I would like to request an alternate explanation of why we cannot have a uniform distribution over integers. If it's the case that your argument does not apply to the Boltzmann distribution, please explain why it does not apply, because I can't see it.

Disinfo's argument looks good to me. Not familiar with the Boltzmann distribution myself, but if it assigns a probability to each integer (and only to the integers), then those probabilities cannot be equal. That's OK, because an infinite sum can converge to a finite value.

Here is an example - if you repeatedly flip a fair coin, what is the probability distribution of N, where N is the first toss that comes up heads? There is a 0.5 probability N=1, a 0.25 probability N=2, a 0.125 probability N=3, and so on. This assigns probability to every integer; it is possible (very unlikely, but possible) that you will get 1,000,000,000,000,000 tosses all tails before you get a head. But:

0.5+0.25+0.125+0.0625+...(and so on forever)

does not add up to infinity. It adds up to a finite number, which is one. So this is a valid probabililty distribution. You just can't make all the probabilities equal. Then the infinite sum wouldn't converge.

Hope this is helpful. You are asking some subtle questions, and you got some bad answers early on :( Disinfo's answers look good to me though.

Disinfo Agent
2007-Sep-11, 10:58 PM
The key is in understanding that what causes trouble with uniform distributions on unbounded sets is the fact that we require all elementary probabilities to be the same number p. Then we run into the fact that p + p + .... is infinite. But this can be circumvented if we allow the individual probabilities to differ, so that their sum may converge. Then we can have

p1 + p2 + p3 + .... = 1

One example is the geometric distribution (http://planetmath.org/encyclopedia/GeometricDistribution2.html). (P.S. which is the example Vadim gave in the post above.)

Another version of the geometric distribution at MathWorld. (http://mathworld.wolfram.com/GeometricDistribution.html)

CodeSlinger
2007-Sep-12, 12:21 AM
*snaps fingers*

Ah, I see! Vadim, DA, those were fantastic explanations. Thank you so much :)

Vadim
2007-Sep-12, 01:10 AM
CodeSlinger, I am glad our posts are helpful. I just want to mention, the last few posts are talking about probability distributions on a countably infinite set, like the integers. When you go to an uncountably infinite set like the real numbers (which is what the first post was about), things are actually a bit qualitatively different, and a lot more subtle. But the Feller and Billingsley books will talk about that.

publius
2007-Sep-12, 01:48 AM
Nice overview of infinite cardinalities here. (http://www.mathacademy.com/pr/minitext/infinity/)
And more (explains Absolute Infinity). (http://dbanach.com/infin.htm)

I see he recommends Rudy Rucker's "Infinity and the Mind" at the end of the first link. That is a good book, which I recommended in the Godel Thread. I would've mentioned it here, but was afraid I was going to sound like some sort of groupie for it. :)

Seriously, it is a good book on this stuff.

-Richard

BioSci
2007-Sep-12, 11:27 PM
Quote:
Originally Posted by BioSci
I think this demonstrates the problem with trying to assign a "probability" or even trying to identify a "random" integer. The problem is that infinty is better thought of as a concept and not a number. In this example "c" can not be a positive number - but it also can not be 0 since according to probability the sum of individual probabilities must sum to 1!

You're right that it cannot be zero in the specific case of the integers. However, it can be zero in other contexts, such as the real numbers.

I do not understand your argument - are you saying that a "random" selection of real numbers can have a different probability of selecting such elements? - how is that random? "randomness" in an infinite set simply can have no real meaning since a meaningful probabilty of any one selection or even any size (less than infinite) set of selected numbers can not be assigned.
If you allow the individual probabilities of selected numbers to differ - you are no longer dealing with a "random" selection of such elements.


Quote:
Originally Posted by BioSci
The question of the probability of randomly selecting a specific number from an infinite set is ill-formed and has no meaning!


Nonsense. Look, how about you guys research things a little before pontificating about them?

Can you give an example of how to select a "random" number (that is, one that has the same chance of being selected as any other number in a set) from an infinite set of numbers and then define the probability of getting that number?
Go ahead - try it - I don't think so! :)


and see here: http://mathforum.org/library/drmath/view/62486.html


The very sentence "1/infinity = 0" has no meaning. Why? Because
"infinity" is a concept, NOT a number. It is a concept that means
"limitlessness." As such, it cannot be used with any mathematical
operators. The symbols of +, -, x, and / are arithmetic operators, and
we can only use them for numbers....
In math, when you hear people say things like "1 over infinity is
zero" what they are usually referring to is something called a limit.
They are just using a kind of shorthand, however. They do NOT mean
that 1 can actually be divided by infinity....
So, to finish up, you are perfectly correct in saying that "1/infinity
= infinitesimally small." But only if you realize that you REALLY mean
"1 divided by a REALLY big number is a REALLY small number."

a1call
2007-Sep-13, 01:37 AM
A word of warning: through out the history there have been people who have gone insane pondering infinity and trying to comprehend it. Accepting the fact that it is beyond our comprehension is wise.
We can get some insight into it's concept by observing the behavior of functions as we substitute consecutively larger (or smaller) values for the variables. This has led to mathematics of calculus and series and the likes. But comprehension (not just proving) that a subset of an unempty set has the same number of elements as the whole set is beyond our real world and is where logic fails and gives illogical conclusions.
As for the original post:
The fact is there can never be a random generator of elements of an infinite set.
Imagine for a second that we invent a machine which upon pressing a bottom generates a random integer in base 10. Then, chances are that the number of digits this integer will have, will be greater than any integer x where x is a finite integer. This is because there are infinitely larger number of integers with more digits than x versus limited number of integers with digits less than or equal to x.
In other words the invented machine will have to put out a number with infinite number of digits which by definition would not be an integer. This is because an integer has to have a finite number of digits. So such a machine can not exist.

a1call
2007-Sep-13, 02:59 AM
Everything we know including the number of neurons we use to know them with is finite.

After having a couple of days to think about this and listening to statements from the Master, I would like to correct the above statement.
The mainstream view of the universe is that of a finite entity. However there are instances where we can deal with infinity in the real world. Surprisingly we are not facing infinity in a large scale but in an infinity small scale. The ancient Greeks came up with the concept of the smallest indivisible constituent of the nature and named it the Atom. Since then we have learned that this element is itself constituted of more fundamental elements and for all we know there is no set smallest indivisible element in nature.
Another brush with infinity would be the infinitely different possible combinations of these elements and compounds which though can not exist naturally in Nature (due to it's finite nature on the large side) can be invented by Humanity.

CodeSlinger
2007-Sep-13, 03:01 AM
While we wait for experienced voices like Vadim and Disinfo Agent to drop in, let's see if I can exercise what little there is of my new-found understanding...

@BioSci: You are correct in that there is no uniform distribution over the infinite set of integers. But you are incorrect in saying that "random" implies that all events must have exactly the same probability. A distribution where the events do not have equal probability (such as the geometric distribution Vadim and DA described) is a perfectly valid random distribution.

@ a1call: I think I spot a flaw in your argument.


Imagine for a second that we invent a machine which upon pressing a bottom generates a random integer in base 10.


In other words the invented machine will have to put out a number with infinite number of digits which by definition would not be an integer.

If the machine will only generate integers, then it will never output a number with infinite number of digits, because as you say, such a number is not an integer, so I don't see how the contradiction will occur.

a1call
2007-Sep-13, 03:15 AM
I believe it's called Transposition (http://en.wikipedia.org/wiki/Transposition_%28logic%29).


If p then q IMPLIES if not q then not p

BioSci
2007-Sep-13, 05:58 AM
@BioSci: You are correct in that there is no uniform distribution over the infinite set of integers. But you are incorrect in saying that "random" implies that all events must have exactly the same probability. A distribution where the events do not have equal probability (such as the geometric distribution Vadim and DA described) is a perfectly valid random distribution.

@ a1call: I think I spot a flaw in your argument


Randomness is actually a rather difficult parameter to measure and there are significant efforts made to develop useful methods for generating random numbers (used for a variety of statistical, mathematical, cryptographic, and scientific tests)

Definitions of random vary from the simple: http://www.atis.org/tg2k/_random_number.html

random number: 1. A number selected from a known set of numbers in such a way that each number in the set has the same probability of occurrence. 2. A number obtained by chance. 3. One of a sequence of numbers considered appropriate for satisfying certain statistical tests or believed to be free from conditions that might bias the result of a calculation.


or more completely, http://mathworld.wolfram.com/RandomNumber.html

A random number is a number chosen as if by chance from some specified distribution such that selection of a large set of these numbers reproduces the underlying distribution. Almost always, such numbers are also required to be independent, so that there are no correlations between successive numbers. Computer-generated random numbers are sometimes called pseudorandom numbers, while the term "random" is reserved for the output of unpredictable physical processes. When used without qualification, the word "random" usually means "random with a uniform distribution."


While one can have random distributions of numbers that correspond to a mathematical distribution (such as Chi-square, normal distribution, F-distribution, etc.) this distribution according to some mathematical formula will not change the problem with trying to identify the "probability" of picking any specific "random" number from such an infinite set of numbers. The fact that the numbers are not linear does not help if you are trying to determine the probability of choosing a member of an infinite set, the "probability" of choosing a specific number has no real meaning because infinity is not a real number. Normal rules of probability simply do not apply when one tries to determine the value of 1/infinity.
What can apply to a distribution is probability of randomly generated numbers falling into set ranges of the distribution. BUT - these randomly generated numbers are not selected from an infinite continuous series but rather from a finite set of random numbers with a specified precision based on the random number generating process.
As soon as you try to introduce infinity into your set from which to choose -- all is lost and you simply can not get there from here!:)

Van Rijn
2007-Sep-13, 09:11 AM
The mainstream view of the universe is that of a finite entity.


The mainstream view is that the observable universe is finite, but that we don't know if the entire universe is infinite or finite.

Here are a couple of threads that have discussions related to this one:

http://www.bautforum.com/questions-answers/62247-what-infinite.html

http://www.bautforum.com/questions-answers/61827-if-universe-infinite.html

Robert TG
2007-Sep-13, 10:13 AM
Your “question about "infinity" in math”

BioSci is correct. Infinity is a concept not a number.

In math it serves the purpose as a concept but has no actual numerical value as a few simple formulas show.

Infinity + 1 = Infinity
Infinity + Infinity = Infinity
Infinity x Infinity = Infinity

These formulas do not make numerical sense because "Infinity is not a number".

So, for a Random Number Generator to pick one number from an infinite number of numbers, the result should produce the highest probability (infinitely high) that the specific number that will be picked will be Infinity ( again the concept). As Infinity is not a number the result is 'meaningless' as BioSci has already said.

Disinfo Agent
2007-Sep-13, 12:44 PM
I do not understand your argument - are you saying that a "random" selection of real numbers can have a different probability of selecting such elements?Yes, of course. For starters, that's what discrete distributions do, innit?

And you mean you've never heard of mixed distributions (http://www.ds.unifi.it/VL/VL_EN/dist/dist3.html)?


If you allow the individual probabilities of selected numbers to differ - you are no longer dealing with a "random" selection of such elements.So the geometric distribution isn't random? :rolleyes:

Go study, BioSci.

CodeSlinger
2007-Sep-13, 02:35 PM
@ a1call: After thinking about it some more in the shower this morning, I think I got it now. Thanks :)

@ Robert TG: Yes, I understand now that one cannot *uniformly* pick a random number out of an infinity of numbers. However, if I have correctly understood the answers given to me in this thread, it *is* certainly possible to randomly pick a number out of an infinity of numbers using a *non-uniform* distribution.

Disinfo Agent
2007-Sep-13, 02:36 PM
@ Robert TG: Yes, I understand now that one cannot *uniformly* pick a random number out of an infinity of numbers. However, if I have correctly understood the answers given to me in this thread, it *is* certainly possible to randomly pick a number out of an infinity of numbers using a *non-uniform* distribution.That's it in a nutshell.

I will add that many laymen confuse "random" with "uniformly random", which causes a lot of confusion. That's because we tend to think of finite spaces only in everyday life. But even in finite spaces one can have non-uniform randomness, like in the case of an unbalanced coin, or a "fixed" roulette.

a1call
2007-Sep-13, 02:56 PM
@ a1call: After thinking about it some more in the shower this morning, I think I got it now. Thanks :)

Well you are definitely in good company:


It is most famously attributed to Archimedes; he reportedly uttered the word when, while bathing, he suddenly understood that the volume of an irregular object could be calculated by finding the volume of water displaced when the object was submerged in water. After making this discovery, he is said to have leapt out of his bathtub and run through the streets of Syracuse naked. :)


Source (http://en.wikipedia.org/wiki/Eureka_(word))

CodeSlinger
2007-Sep-13, 03:02 PM
One key difference between Archimedes and me: I am definitely not ballsy enough to run naked through streets of my neighborhood :)

BioSci
2007-Sep-13, 05:26 PM
That's it in a nutshell.

I will add that many laymen confuse "random" with "uniformly random", which causes a lot of confusion. That's because we tend to think of finite spaces only in everyday life. But even in finite spaces one can have non-uniform randomness, like in the case of an unbalanced coin, or a "fixed" roulette.

But the problem is that if you are selecting from a continuous, infinite set of such numbers it dosen't matter how they are grouped or if they are uniformally random. Unless you constrain the precission of the data set it will contain an infinite number of such points and the "probability" of selecting any one such point is still 1/infinity which is not defined. Probability can only be calculated and only is meaningful if one is looking at a range of continuous values or a finite set from which to choose. Picking a specified point from a continuous, non-uniform (infinite) distribution or a specific number from a uniform infinite set are equally meaningless in terms of probability.

Perhaps you are confusing the probability of a random point falling into a geometric region: such as 1/2, 1/4, 1/8, ...? such a series does have an infinite number of elements - but you are not selecting the individual elements of the infinite set randomly - rather you are using the numeric value of your set elements as a range to determine probability.

Disinfo Agent
2007-Sep-13, 05:31 PM
But the problem is that if you are selecting from a continuous, infinite set of such numbers it dosen't matter how they are grouped or if they are uniformally random. Unless you constrain the precission of the data set it will contain an infinite number of such points and the "probability" of selecting any one such point is still 1/infinity which is not defined.The probability is not necessarily zero, because you can do the random choice in such a way that some reals occur (much) more often than others. Any discrete distribution effectively does that, but there are other possibilities, such as the mixed distributions I mentioned above, which assign a positive probability to a few arbitrary reals, while being continuous elsewhere on the real line.


Probability can only be calculated and only is meaningful if one is looking at a range of continuous values or a finite set from which to choose.In your opinion. The folks who have studied and built probability theory have something different to say.


Picking a specified point from a continuous, non-uniform (infinite) distribution or a specific number from a uniform infinite set are equally meaningless in terms of probability.You use the word "meaningless" rather liberally. I might even say "meaninglessly". Any solid book on probability theory will include a careful discussion of the notion of random experiment, and make it clear that random is not the same as uniformly random, in probability theory.

BioSci
2007-Sep-13, 06:34 PM
The probability is not necessarily 1/infinity, because you can do the random choice in such a way that some reals occur (much) more often than others. Any discrete distribution effectively does that, but there are other possibilities, such as the mixed distributions I mentioned above, which assign a positive probability to a few arbitrary reals, while being continuous elsewhere on the real line.

Yes, you can define probability of obtainng a "few arbitrary reals" differently in a mixed distribution - but that is a different problem and stretches the meaning of random selection of set members.
You no longer are asking the probabilty of randomly selecting a specific number from an infinite or continuous set but rather (say): "if 1/2 the time I select 10.1, what is the probability of selecting 10.1 from the set that contains 10.1 and the set of integers (perhaps it alternates 10.1 and integers?)"

The issue of mixed distrubutions or non-uniform probability do not change the original question - if one selects from a continuous/integer infinite population, unless the selection critria somehow defines the probability of a particular selection (through bias or other non-random criteria); or certain numbers are repeated infinitely (see above); the probability of any specific number being selected becomes 1/infinity.


For example, review your statistics/probability basics, one can not calculate the probability of getting a specific value on a continuous normal curve - it must be a range of values (or implicit in the calculation is a +/- value for rounding).

Disinfo Agent
2007-Sep-13, 07:13 PM
You no longer are asking the probabilty of randomly selecting a specific number from an infinite or continuous set but rather (say): "if 1/2 the time I select 10.1, what is the probability of selecting 10.1 from the set that contains 10.1 and the set of integers (perhaps it alternates 10.1 and integers?)"I don't see where's the difference. The question which started all this was, basically, "If p x 100% of the time I choose an integer at random, which value may I assign to p?"

The answer is, either you allow different 'p's for different integers, or you can't do that.

And an extension: "If p x 100% of the time I choose a real number at random, what should p be?"

The answer now is that p must be zero for almost all reals (it may be positive for a few of them -- this leads to a mixed distribution).


For example, review your statistics/probability basics, one can not calculate the probability of getting a specific value on a continuous normal curve - it must be a range of values (or implicit in the calculation is a +/- value for rounding).Oh, yes you can calculate that probability; it's zero. Check it up in the various links I have posted in this thread so far.

BioSci
2007-Sep-13, 10:47 PM
Oh, yes you can calculate that probability; it's zero. Check it up in the various links I have posted in this thread so far.

Yes, one can "calculate" it to be zero but now tell me what that means if every value gives the same probability (0) - it doesn't matter if I ask for a point close to the mean or 100 sigma away - the probability is "calculated" to be the same. Does that mean that none of those values can be selected at random as a function of the normal distribution (probability = zero)?

Such a probability in a real sense is meaningless! (except to tell one that they asked a useless question!)

What does it mean if the probability of obtaining any and all specific values - even if clearly contained in the main area of a normal distribution - is zero? (Hint: perhaps it means one asked a "bad" question)

this whole question is frought with undefined math:
0 x infinity =?
1/infinity =?
1/infinity x infinity = ?
infinity/infinity = ?


I don't see where's the difference. The question which started all this was, basically, "If p x 100% of the time I choose an integer at random, which value may I assign to p?"

The answer is, either you allow different 'p's for different integers, or you can't do that.

And an extension: "If p x 100% of the time I choose a real number at random, what should p be?"

The answer now is that p must be zero for almost all reals (it may be positive for a few of them -- this leads to a mixed distribution).


The difference is that it does not matter if one selects from integers or reals, if you select from an infinite set of either group no meaningful probability can be assigned to a specific selection - unless you rig the set to include an infinite repetition of certain numbers but any other numbers that are not repeated an infinite number of times then turn out to have a zero "possibilty" of being selected. You only changed the game for a limited set of numbers but have not altered the problem with selecting from the remaining finite numbers in the infinite set. It doesn't matter how the "density" of your numbers is arranged - all specific (noninfinitely repeated) numbers will have the same undefined "probability" of being selected - 1/infinity. This is also true even if you repeat certain numbers any finite number of times - they will be just as impossible to identify a meaningful probability of selecting as a single number in the background sea of infinite numbers to choose from.

In the example I gave, one has only shifted the "probability" problem from 0 +0 + 0 ... = 1? to 1/2 + 0 + 0 +0 ... = 1?

The real answer is that you simply can not assign a meaningful value to the value of P. It doesn't matter if one is selecting from integers or reals and altering the density of reals does not help, you still can not change the value of P for any specific number unless you specify that your infinite number set contains that number an infinite number of times! (Then the probability of that number being selected depends on how you constructed your infinite set!)

Disinfo Agent
2007-Sep-13, 11:02 PM
[...] in a real sense is meaningless [...] whole question is frought with undefined math [...] undefined "probability" [...] impossible to identify a meaningful probability [...] The real answer is [...]I'm not wasting any more time with you until you get off that high horse of yours.

Here's my last piece of free advice to you: go learn some probability. Now.

Kullat Nunu
2007-Sep-17, 03:23 PM
Probability of getting anything in an infinite set is:
1/infinity
Which is undefined, not 0.

Well, use limes if you want to be accurate. 1/x = 0 is certainly true when x → ∞.


It is the same as divide by 0.

No, it is not. 1/x goes to -∞ when x- → 0 and ∞ when x+ → 0. Therefore 1/0 is undefined. However, using 0 as a denominator is legal if the discrepancy is solved: |1/x| = ∞ when x → 0, or when x is defined only either in [0, ∞[ or ]-∞, 0]. These cases are similar to 1/∞.

a1call
2007-Sep-17, 03:44 PM
Well, use limes if you want to be accurate. 1/x = 0 is certainly true when x → ∞.


Not really, Mathematically speaking 1/x converges to 0 as x approaches ∞.

It can not be equal to 0 as explained in the link which I had provided and is being included again:

here (http://mathforum.org/library/drmath/view/62486.html)

divide by 0 is undefined as is 1/∞:

From the above link:


To write 1/infinity and mean "1 divided by infinity" doesn't make any
sense
In other words it is undefined



No, it is not. 1/x goes to -∞ when x- → 0 and ∞ when x+ → 0. Therefore 1/0 is undefined. However, using 0 as a denominator is legal if the discrepancy is solved: |1/x| = ∞ when x → 0, or when x is defined only either in [0, ∞[ or ]-∞, 0]. These cases are similar to 1/∞.


What exactly do you mean by x+ → 0 and x- → 0?

Kullat Nunu
2007-Sep-17, 04:30 PM
Not really, Mathematically speaking 1/x converges to 0 as x approaches ∞.

That's exactly what x → ∞ means: limx → ∞ 1/x = 0 is true.


divide by 0 is undefined as is 1/∞:

Well, 1/∞ converges but 1/0 does not. 1/∞ is "infinitely far away", 1/0 is simply not defined. The function 1/x reaches 0 at infinity, but never 1/0.

You and others here are right in that ∞ and thus 1/∞ are not values in the normal sense. I'm thinking ∞ as a limit (infinite limit?! :)), not as a fixed value which it is not. In this sense 1/∞ = 0 and ∞ * ∞ = ∞ and other "weird" equations are correct.


What exactly do you mean by x+ → 0 and x- → 0?

x+ → 0 means we close in on 0 from the positive side (x > 0) and x- → 0 opposite of that. In the former case the function 1/x goes up rapidly, and in the latter case it drops. If limx+ → b f(x) → a and limx- → b f(x) → a the function f(x) is defined at that point. It is obviously not true in the case of f(x) = 1/x, x → 0, but for example f(x) = |1/x| is defined at 0.

Kullat Nunu
2007-Sep-17, 09:39 PM
I see he recommends Rudy Rucker's "Infinity and the Mind" at the end of the first link. That is a good book, which I recommended in the Godel Thread. I would've mentioned it here, but was afraid I was going to sound like some sort of groupie for it. :)

Seriously, it is a good book on this stuff.

-Richard

It's dangerous for mental health. :shifty:

publius
2007-Sep-17, 10:04 PM
It's dangerous for mental health. :shifty:

Hey, that horse has been out of the barn with me for a long time before I started worrying about infinity, so no problem. :lol:

-Richard

a1call
2007-Sep-18, 02:02 PM
The debate into if 1/∞ does equal 0 or not is a bit illogical. There are statements for and against it from same reputable mathematical sources. It is very subjective to what definition and convention you are using. due to the funny nature of the concept of infinity where there are indeterminate forms (http://mathforum.org/library/drmath/view/55764.html). Still I would argue (and I am not the only one) that 1/∞ does not equal 0.
Firstly infinity is not a number but a concept and as explained previously 1/∞ is meaningless and can not equal a number yet alone 0.
If you accept the following:

In a more down-to-earth sense, the words "approaches infinity" are used in place of the words "increases without limit." Thus, it is said that the limit of 1/x, as x approaches infinity, is equal to zero. In this context, infinity does not represent a defined quantity, but is merely a convenient expression.

Source (http://whatis.techtarget.com/definition/0,,sid9_gci809150,00.html)

Then 1/∞ is not a number and then it can not be equal to 0.

Even if we assume that infinity is a real or complex number (which it is not) it can still be "proved" that 1/∞ is not 0. Here is how.

1/∞ would be of the form x/y where x and y are real (or complex which includes real numbers) numbers. So we can write:

x/y=0 => x=0*y => x=0
for any (and all :)) x and y where x and y are real numbers and y<>0 (Read y not equal to 0).
This would imply that if 1/∞=0 then 1=0 which is false.
Again using Transposition (http://en.wikipedia.org/wiki/Transposition_%28logic%29)

we can conclude that the statement
1/∞ = 0
is false. For further clarity "Transposition" is the reasoning where you would say:

If Statement A is True then Statement B is True. If/Since Statement B is False then Statement A is False.

Here:

Statement A: 1/∞ = 0
Statement B: 1=0

Of Course due to the funny nature if infinity it is as easy to prove the exact opposite. But I favor this conclusion as IMHO it complies with the proper definition if Infinity and division.

mugaliens
2007-Sep-18, 05:20 PM
This fallacy prevents peole using "1 2 3 4 5 6" as their lottery number, when the odds of that coming up are exactly the same as any other number.

John

True.

However, if you avoid all possible combinations of birthdays and other stuff people commonly use when picking lottery numbers, you can significantly decrease the liklihood that you'll have to share the winning prize with others.

The fallacy is that the rate of return of most lottery systems is decidely less than 1.0. It's more like .50.

I get 1.045 at the bank, about a few times more playing the markets, which makes it ridiculous for me to spend money on Lotto tickets.

Disinfo Agent
2007-Sep-18, 05:21 PM
Firstly infinity is not a number but a concept [...]I often hear that mantra, but I've never quite understood it. Care to define "number" for me?...

Theorem: 1/infinity = 0.

Proof:


Let (sn) be an arbitrary sequence of real numbers such that sn --> +infinity. Then, it's easily shown that the sequence 1/sn converges to 0. Thus, by definition, 1/infinity = 1/(limsn) = lim(1/sn) = 0. QED.

BioSci
2007-Sep-18, 08:10 PM
I often hear that mantra, but I've never quite understood it. Care to define "number" for me?...


Start here for infinty then look at most definitions of numbers (reals) and inifinity to see how they have to be treated differently.

http://mathforum.org/library/drmath/view/55765.html

"One of the main reasons we don't call infinity a number is that it doesn't
act like one."

and: http://mathforum.org/library/drmath/view/62486.html

"Where did you get the idea that 1/infinity = 0?
The very sentence "1/infinity = 0" has no meaning."

Disinfo Agent
2007-Sep-18, 08:20 PM
http://mathforum.org/library/drmath/view/55765.html

"One of the main reasons we don't call infinity a number is that it doesn't
act like one."The reply in there concerns infinite cardinalities (set theory). The conversation here is about calculus.


and: http://mathforum.org/library/drmath/view/62486.html

"Where did you get the idea that 1/infinity = 0?

The very sentence "1/infinity = 0" has no meaning."Rubbish. Doctor Wallace apparently has never picked up a book on elementary calculus. :rolleyes:

P.S. 1/infinity can be defined as the limit of 1/sn, where (sn) is an arbitrary sequence of real numbers such that sn --> infinity.