Undergrad How Can Improbability and Infinitesimal Probabilities Exist in Real Life Events?

[WWGD]

Could you please elaborate? Did you mean to use the open interval? What do you mean by "the same"? Are you agreeing with my claim? I was merely distinguishing between the maximally improbable, and the impossible, and claiming that only the latter lel of probability should be called zero.

Re: 'the same', I mean that for any x in (0,1), sorry( any interval using the uniform distribution will do) , there is no s>0 with P((x-s, x+s)) =0 . This is one possible distinction of impossibility with probability 0 that can be used with an interval. Please give me some time, I will address the other point.

 

[Stephen Tashi]

There is in fact a mathematical theory which fully deals with these issues called possibility theory and which answers many questions which probability theory simply can not, while being consistent with probability theory namely by having probability theory itself as a special limiting case.

Yes, but I've see nothing in possibility theory that deals with random sampling. It's conceivable that one might apply possibility theory to specific cases of random sampling, but this is, again, a situation where a mathematical theory does not comment on a particular phenomena. It is up to those who apply the theory to decide what is possible or impossible in the phenomena.

The metaphysics of possibility and impossibility are interesting to discuss. There is also the metaphysical question of how the possibility of an event relates to the actual occurence of an even. (E.g. can an event be "possible" but never actually happen?) It's an interesting metaphysical question of whether there can exist a continuous random variable that can be sampled exactly. There is the metaphysical question of how to regard sets that are not lebesgue measureable and cannot be assigned a probability (even a probability of zero) by a probability distribution such as the uniform distribution on [0,1] - is it impossible or possible to select a random sample that is a member of such a set?

Nothing in probability theory answers these metaphysical questions. Of course this doesn't stop experts in probability theory from stating their own personal opinions about such matters.

 

[sysprog]

@Stephen Tashi My intention in that diatribe was to say in English that to make such a claim as that a positive number, however small it may be, can be zero, is inconsistent, and is incorrect use of language.

Re: 'the same', I mean that for any x in (0,1), sorry( any interval using the uniform distribution will do) , there is no s>0 with P((x-s, x+s)) =0 . This is one possible distinction of impossibility with probability 0 that can be used with an interval. Please give me some time, I will address the other point.

Sure, thanks, @WWGD

 

[WWGD]

@Stephen Tashi My intention in that diatribe was to say in English that to make such a claim as that a positive number, however small it may be, can be zero, is inconsistent, and is incorrect use of language.

Sure, thanks, @WWGD

The Archimedean property of the standard Reals dictates that a number that is indefinitely-small must equal zero. In the non-standard Reals, this property does not hold and you can have infinitesimals and maybe there is some way of having infinitesimal-valued probabilities but I am not aware of any. The properties of the standard Reals do not allow for the assignment of non-zero Real probabilities to more than countably-many points, as a sum with uncountable support ( meaning uncountably-many non-zero terms) will not converge, let alone add up to 1.

 

[sysprog]

The Archimedean property of the standard Reals dictates that a number that is indefinitely-small must equal zero. In the non-standard Reals, this property does not hold and you can have infinitesimals and maybe there is some way of having infinitesimal-valued probabilities but I am not aware of any. The properties of the standard Reals do not allow for the assignment of non-zero Real probabilities to more than countably-many points, as a sum with uncountable support ( meaning uncountably-many non-zero terms) will not converge, let alone add up to 1.

Even allowing for that, there is a difference between an indefinitely small probability and an outright impossibility. I'm really criticizing the incorrect use of language. I think it's logically not acceptable to say that something is both non-zero and zero.

 

[FactChecker]

Even allowing for that, there is a difference between an indefinitely small probability and an outright impossibility. I'm really criticizing the incorrect use of language. I think it's logically not acceptable to say that something is both non-zero and zero.

It's more than a language problem. Suppose a point on the [0,1] line segment is randomly selected from a uniform distribution. A point WAS selected, yet there was a zero probability of that point having been selected. It was possible with an infinitely small probability. That is a difficulty of probability, not of the language. On the other hand, there are examples of truly impossible things, like obtaining a 10 from the roll of a single normal die.

 

[sysprog]

It's more than a language problem. Suppose a point on the [0,1] line segment is randomly selected from a uniform distribution. A point WAS selected, yet there was a zero probability of that point having been selected. It was possible with an infinitely small probability. That is a difficulty of probability, not of the language. On the other hand, there are examples of truly impossible things, like obtaining a 10 from the roll of a single normal die.

I firmly reject the complacent use of incorrect language as an expedient in the matter.

 

[WWGD]

I firmly reject the complacent use of incorrect language as an expedient in the matter.

Still, however imperfect, term overloading is arguably better than other alternatives. If you were to use absolutely precise and unambiguous terminology it would be essentially impossible to understand when you spoke and people would be upset. (Over) simplification and a certain level of ambiguity seem like necessary evils and used in most, if not all technical areas.

 

[FactChecker]

I firmly reject the complacent use of incorrect language as an expedient in the matter.

Sorry, I think I read too much into your prior post. Certainly, when this specific subject is being discussed, some clear language using different terms is practical and helpful.

That being said, I think that this subject is not usually an issue and using different terms in general would just be confusing and unnecessary.

 

[WWGD]

Dont get me wrong @sysprog , it is ambiguous and confusing but it is too difficult to be meticulously precise about very technical topics and most of the time it will not happen and the best I can rhink of doing is asking for clarification.

 

[PeroK]

It's more than a language problem. Suppose a point on the [0,1] line segment is randomly selected from a uniform distribution. A point WAS selected, yet there was a zero probability of that point having been selected. It was possible with an infinitely small probability. That is a difficulty of probability, not of the language. On the other hand, there are examples of truly impossible things, like obtaining a 10 from the roll of a single normal die.

It's only possible to select from a finite set with a uniform distribution; or from a coutable set with a non uniform distribution. It's not possible to devise an algorithm that could select from an include set - in the sense that the algorithm has an uncountable number of possible outputs.

PS you could have a random variable uniformly distributed on an uncountable set. But that is something else entirely. It's the same difference as defining an infinite sine function and supposing you have physically drawn an infinite sine function.

 

[Stephen Tashi]

It's only possible to select from a finite set with a uniform distribution; or from a coutable set with a non uniform distribution.

You could modify that statement to involve only the mathematical properties of probability distributions and stay within the domain of probability theory. Once you begin to speak of the possibility of taking random samples, you are wandering outside the scope of probability theory.

To repeat, the theory of probability says nothing about the possibility or impossibility of selecting random values from a distribution. The discussion of whether algorithms exist to do this falls under the heading of theories of computability or some other field of science or mathematics.

In particular, the question of whether algorithms exist that can take random samples is a narrower question than whether physical processes exist that do this. For example, if the time for an atom to decay is actually given by an exponential distribution then Nature can can sample from a continuous distribution, even if human beings can only measure the time of decay with finite precision. Whether it is possible for Nature to do this is a topic in physics. It is not covered by probability theory.

 

[FactChecker]

It's only possible to select from a finite set with a uniform distribution; or from a coutable set with a non uniform distribution. It's not possible to devise an algorithm that could select from an include set - in the sense that the algorithm has an uncountable number of possible outputs.

I am uncomfortable with that statement. It strikes me as confusing a human inability to define a process with the claim that no such thing exists. Sort of like claiming that there is no such thing as the area of a circle because there is no way to square a circle.

 

[PeroK]

I am uncomfortable with that statement. It strikes me as confusing a human inability to define a process with the claim that no such thing exists. Sort of like claiming that there is no such thing as the area of a circle because there is no way to square a circle.

A circle and the area of a circle are well defined mathematically. And loosely one can draw a circle. But, you shouldn't confuse the two.

The problem with your paradox that the impossible can happen is that it confuses real and mathematical processes.

 

[FactChecker]

The problem with your paradox that the impossible can happen is that it confuses real and mathematical processes.

I don't agree that I am confusing real and mathematical processes. I am trying to stay within the confines of mathematical definitions (not processes). I would leave the process of selection undefined and assume that any real number that exists can be selected somehow (maybe by a "god-like" process). I think this is an important difference from one which says that only a countable set can be selected from.

I admit that your position has a great advantage if one states that a selection must be done by some definable process. That does seem reasonable. Is there some body of work that addresses this issue, which you are basing your position on? I admit that I have never looked into it.

 

[PeroK]

I don't agree that I am confusing real and mathematical processes. I am trying to stay within the confines of mathematical definitions (not processes). I would leave the process of selection undefined and assume that any real number that exists can be selected somehow (maybe by a "god-like" process). I think this is an important difference from one which says that only a countable set can be selected from.

I admit that your position has a great advantage if one states that a selection must be done by some definable process. That does seem reasonable. Is there some body of work that addresses this issue, which you are basing your position on? I admit that I have never looked into it.

The related issue is that only a countable subset of the real numbers are computable. So, the real numbers generally cannot be selected and processed at all!

There's plenty of reference material on that.

The issue that one cannot have a uniform selection process on the natural numbers is well known. There must be reference on that.

You could look for something on the $[0,1]$ paradox. I don't remember what I found last time.

I'll have a look when I get the chance.

 

[FactChecker]

So, the real numbers generally cannot be selected and processed at all!

Is this somehow a denial of the Axiom of Choice?

 

[PeroK]

Is this somehow a denial of the Axiom of Choice?

No. But highlights the difference between sets of numbers you can study using mathematics and numbers that you can select, describe and process.

Look up "computable" numbers.

 

[FactChecker]

@PeroK , Suppose I define a selection process as follows:
I let you define a selection process on the [0,1] line segment that I have no knowledge of or influence on. Let $P$ denote the countable set of possible results of your process and $I$ denote the remainder of [0,1] of numbers that are impossible to select using your process. $P$ has measure zero and $I$ has measure 1. If I apply the Axiom of Choice to claim a chosen value $c$ from $I$, I must say that it had probability zero, even though it was selected.

 

[FactChecker]

Look up "computable" numbers.

This may be the crux of the matter. "computable" implies a finite, terminating algorithm. I like to think of probability of selection as including infinite, "god-like", selection processes. The limitations of humans to compute a number are not always applicable. But I am afraid that I am taking this into a philosophical turn that is not appropriate in this forum. I will look at the subject material that you suggested.

 

[PeroK]

@PeroK , Suppose I define a selection process as follows:
I let you define a selection process on the [0,1] line segment that I have no knowledge of or influence on. Let $P$ denote the countable set of possible results of your process and $I$ denote the remainder of [0,1] of numbers that are impossible to select using your process. $P$ has measure zero and $I$ has measure 1. If I apply the Axiom of Choice to claim a chosen value $c$ from $I$, I must say that it had probability zero, even though it was selected.

Let me describe the issue as follows. You have a real number lottery. Everyone gets to choose their own real number, say, and put it in a sealed envelope. You choose the winning number by whatever process you like. But, you must publish an actual number.

You are not allowed to say you picked "some" number $c$, but you don't know what it is. Nor can you describe it in any way.

Then you are limited to the computable numbers.

It's nothing to do with the axiom of choice.

 

[Quasimodo]

I think the answer to the OP's original question : "how improbable is impossible?" depends on the size of the sample space of the experiment used to derive the probabilities.

I think everyone here would agree, at first sight, that the probability of a random number generator ( producing 0-9 digits one at a time ) to output an infinite string of all 0's is 0 itself ( an impossible outcome. )
And yet there is a sample space where this probability is 1.

 

[PeroK]

I think the answer to the OP's original question : "how improbable is impossible?" depends on the size of the sample space of the experiment used to derive the probabilities.

I think everyone here would agree, at first sight, that the probability of a random number generator ( producing 0-9 digits one at a time ) to output an infinite string of all 0's is 0 itself ( an impossible outcome. )
And yet there is a sample space where this probability is 1.

A random number generator can only ever produce a finite sequence of digits.

 

[Quasimodo]

A random number generator can only ever produce a finite sequence of digits.

Let us please not argue for argument's sake, and accept that there is a true random number generator somewhere producing one digit 0-9 at a time forever, ok?

 

[PeroK]

Let us please not argue for argument's sake, and accept that there is a true random number generator somewhere producing one digit 0-9 at a time forever, ok?

It can produce numbers for an indefinite period, if you like, but it never produces an infinite sequence.

To get an infinite sequence you have to appeal directly to mathematics.

Let $s_n$ be an infinite sequence of digits, where each digit is uniformly distributed on $0-9$, is perfectly valid.

Saying that such a sequence could come from a random number generator is a confusion of mathematical and computational ideas.

 

[FactChecker]

A random number generator can only ever produce a finite sequence of digits.

Unless the first digit takes 1/2 sec, the second digit takes 1/4 sec, the third digit takes 1/8 digit, etc. I think that your logic and objections are based on physical constraints that are not applicable in all the theoretical and conceptual situations that probabilities can reasonably be applied to.

 

[Quasimodo]

Let snsns_n be an infinite sequence of digits, where each digit is uniformly distributed on 0−90−90-9, is perfectly valid.

ἔστω:
so be it, if you like!

 

[PeroK]

Unless the first digit takes 1/2 sec, the second digit takes 1/4 sec, the third digit takes 1/8 digit, etc. I think that your logic and objections are based on physical constraints that are not applicable in all the theoretical and conceptual situations that probabilities can reasonably be applied to.

It's a good point. Then we see precisely the reason that the "impossible" has happened.

1) we postulate a random number generator according to your specification.

2) it generates an infinite sequence in one second.

3) the probability that that precise sequence would be generated is zero.

4) the impossible has happened.

But, we have postulated a physically impossible random number generator. So, no mystery and no paradox. An impossible machine has done the impossible!

 

[FactChecker]

But, we have postulated a physically impossible random number generator. So, no mystery and no paradox. An impossible machine has done the impossible!

Impossible physically or impossible conceptually? In the real world, it is not possible to have an absolutely fair coin, so should we stop talking about the probabilities of a fair coin?

 

[Quasimodo]

Please read my post carefully!

I said, that we can show that there exists a sample space where this probability is 1 and NOT 0!

The proof relies on limits at infinity, so my previous example is realistically viable but if you want to argue trivialities with me, I might as well leave this conversation...