01 Feb 16
Originally posted by twhiteheadTHIS is where we disagree! And I don't care whether you can or cannot define the impossible event as being "in a 'sample space' ". So it isn't in and isn't allowed in a/any sample space; I say we STILL can say it has exactly zero probability!
... If it is impossible then the probability is undefined. ...
I think I, at last, now have all the necessary info to decide what exactly will be my next question to that maths expert in statistics and I will then get back to on that later after he answers.
01 Feb 16
1 edit
Originally posted by humyBut would you mean anything meaningful when you said it? Certainly you would not mean anything mathematical. It is possible that English is less rigorous in its definition of the word, but you shouldn't confuse the two (English and mathematics).
I say we STILL can say it has exactly zero probability!
I think if you actually asked people: what are the chances that a worm could burrow to the centre of the earth, some would say 'no chance' and others would try to change the question by answering 'its impossible'. I favour the latter as assigning zero probability essentially admits possibility.
The question itself could be said to be incoherent as you are asking 'how frequently does something impossible happen?'
01 Feb 16
I have given this a lot of thought. There is clearly something fundamentally different between saying something cannot happen and the question of how likely something is. But I am undecided as to whether or not saying something has no chance of happening is a reasonable way to express the fact that it is impossible. Are there, for example calculations in which such an assignment would make things easier? If you have a two headed coin and a normal coin and you flip both, do you give the two headed coin flip a sample space that includes heads and tails but assign a zero probability to the tails?
So if you flip both coins you first say either flip could be heads or tails and proceed from there?
01 Feb 16
Originally posted by twhiteheadPossibly this is why people distinguish between domain and support. Suppose you have some system with time dependent probabilities. Then points in the sample space will have zero probability at one time but non-zero probability at other times. I think most people would regard them as viable members of the sample space at all times.
No. That would violate the definition of 'sample space'.
Since your expert seems to have a very low opinion of Wikipedia (which I think is unjustified and elitists and has no place in mathematics), let me give a reference to a document from the University of Illinois. I have not checked the credentials of said university.
http://www.math.uiuc.edu/~k ...[text shortened]... f exactly 'zero' would imply it is not a possible outcome and therefore not in the sample space.
01 Feb 16
Originally posted by DeepThoughtIt would appear that the OP situation had a domain is an open set starting at zero (excluding zero) and the support is a closed set starting at zero (and including zero).
Possibly this is why people distinguish between domain and support. Suppose you have some system with time dependent probabilities. Then points in the sample space will have zero probability at one time but non-zero probability at other times. I think most people would regard them as viable members of the sample space at all times.
The value -1 is outside both the domain and the support.
https://en.wikipedia.org/wiki/Support_(mathematics)
Although Wikipedia suggests it may be more relaxed in probability theory.
01 Feb 16
Originally posted by DeepThoughtLebesgue integration is done over measurable sets, which in Euclidean space normally includes the open sets constituting the "standard topology".
You need to be very careful with that kind of detail in Wikipedia. I'd not heard of proper and improper integrals before, and it may be that the Wiki writer is relying on a single reference and the term isn't widely accepted. I'm a bit skeptical about it for two reasons. The first is that Lebesgue integration is all defined in terms of open sets - alt ...[text shortened]... asure set, so including and excluding the boundary points won't make a difference to the result.
01 Feb 16
9 edits
twhitehead
I asked Clyde Oliver, a maths expert on Probability & Statistics,
at: http://www.allexperts.com/cl2/8/science/Mathematics/
about if, in conventional terminology of statistics, something that is impossible has a probability of zero or is undefined.
I also asked him that in the form of my 'earthworm' question.
+ I asked some related stuff about events and sample space.
This is his exact answer, which is very short and you can directly view at:
http://www.allexperts.com/user.cgi?m=6&catID=2077&expID=124031&qID=5089062:
Events that are impossible have probability zero. Always. There is no requirement that events in a sample space have nonzero probability.
Which is exactly as I thought.
So the probability of, say, an earthworm doing something impossible (such as going to the center of the earth intact which I presume really is completely impossible) is defined and is defined as zero and, depending on how you define your setup, that impossible event could be allowed in your sample space. (not sure I can strictly correctly say that as just "be in your sample space" without possibility of misunderstanding or that I must say it as "be allowed in your sample space". Anyone? )
01 Feb 16
Originally posted by humyThat would seem to contradict every definition I have been able to find of 'sample space'. I would be interested if you can find a definition compatible with that claim.
Quoting Clyde Oliver:
There is no requirement that events in a sample space have nonzero probability.
01 Feb 16
1 edit
Here are some lecture notes from a university:
http://cseweb.ucsd.edu/~dasgupta/103/1.pdf
A text book:
https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/amsbook.mac.pdf
A text book written by MIT lecturers:
http://www.athenasc.com/Ch1.pdf
All three of the above are very clear that a sample space is the set of all possible outcomes. There is no hint that impossible outcomes may be included and in fact the definitions almost certainly should be read as being exclusive.
So I am not just relying on Wikipedia.
The only wriggle room I see is to say that all points internal to the earth are possible places a worm could wriggle to, but some of the points are impossible to get to. I'm trying very hard not to cause a contradiction, but 'possible' and 'impossible' seem like opposites to me.
01 Feb 16
4 edits
Originally posted by twhiteheadI believe he meant that as:
That would seem to contradict every definition I have been able to find of 'sample space'. I would be interested if you can find a definition compatible with that claim.
There is no requirement that events allowed in a sample space have nonzero probability.
OBVIOUSLY, he wasn't trying to say you could actually observe impossible events/outcomes occurring in an actual sample space. SURELY you should have figured that out for yourself?
And what do you say of his other comment that I ( + my two brothers which I now asked ) agree with which is:
"Events that are impossible have probability zero. Always."
?
Your earlier assertions clearly contradicted that.
01 Feb 16
7 edits
Originally posted by twhitehead
Here are some lecture notes from a university:
http://cseweb.ucsd.edu/~dasgupta/103/1.pdf
A text book:
https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/amsbook.mac.pdf
A text book written by MIT lecturers:
http://www.athenasc.com/Ch1.pdf
All three of the above are very clear that a sample space is the set of all ...[text shortened]... ry hard not to cause a contradiction, but 'possible' and 'impossible' seem like opposites to me.
All three of the above are very clear that a sample space is the set of all possible outcomes.
In philosophy as here, there are two different KINDS of 'possible '. There is 'causally possible' and 'logically possible'.
Something can be both 'logically possible', which means there is no logical contradiction in it being true, AND yet NOT 'causally possible', where 'causally possible' means it is possible according to natural law that exists in the actual external world.
Note what is 'logically possible' takes no account of natural law (the laws of physics, chemistry etc. ) because there is no logical contradiction in having the universe with natural law being all different. Thus an earthworm traveling intact to the center of the Earth is causally possible but NOT logically possible since it might be logically possible if all natural laws where different.
So, how do you know that what they mean by "possible outcomes" in the sample space is, exactly what I always have believed they mean, "causally possible outcomes" and NOT "logical possible outcomes" thus allow for the former kind of 'impossible' outcomes to be in the sample space?
01 Feb 16
Originally posted by humyThere's a slight disagreement I have there. It is possible (in the sense of not ruled out) that the laws of physics are not contingent, in the sense that any fundamental theory other than the "one true" theory of everything, has some internal contradiction. This would obviate the need for empirical testing of the theory, provided it could be shown that no other theory is non-self-contradicting. If that is the case the only variation between possible worlds in the laws of physics would be things like the electro-weak mixing angle. I suspect that there is more than one way things could be, but that caveat does exist.All three of the above are very clear that a sample space is the set of all possible outcomes.
In philosophy as here, there are two different KINDS of 'possible '. There is 'causally possible' and 'logically possible'.
Something can be both 'logically possible', which means there is no logical contradiction in it being true, AND y ...[text shortened]... ble outcomes" thus allow for the former kind of 'impossible' outcomes to be in the sample space?
01 Feb 16
5 edits
Originally posted by DeepThoughtYes. I find that comment interesting because I had once thought along very similar lines!
There's a slight disagreement I have there. It is possible (in the sense of not ruled out) that the laws of physics are not contingent, in the sense that any fundamental theory other than the "one true" theory of everything, has some internal contradiction. [i]This would obviate the need for empirical testing of the theory, provided it could be shown t ...[text shortened]... g angle. I suspect that there is more than one way things could be, but that caveat does exist.
I wondered if perhaps the is some unknown purely deductive reason why the laws of nature couldn't possibly be any different!? If so, I wonder if that reason is so incredibly subtle that it would always be beyond our comprehension?
perhaps I shouldn't have said;
"there is no logical contradiction in having the universe with natural law being all different. "
But rather;
"there is no reason that we know of why we cannot have natural law of the universe being different in a self-consistent way i.e. still be without logical contradiction. "
02 Feb 16
Originally posted by humyI think one would need some way of dismissing all other theories. It's difficult to see how that could be done without having some means of automatically generating theories and showing that they contain contradictions.
Yes. I find that comment interesting because I had once thought along very similar lines!
I wondered if perhaps the is some unknown purely deductive reason why the laws of nature couldn't possibly be any different!? If so, I wonder if that reason is so incredibly subtle that it would always be beyond our comprehension?
perhaps I shouldn't have said; ...[text shortened]... universe being different in a self-consistent way i.e. still be without logical contradiction. "