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Old 16th April 2012, 11:57 AM   #41
Vorpal
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Originally Posted by eijah View Post
As I proposed earlier, and as someone else here said about either false or true, it seems to me rightly or wrongly that something is either not possible 0 or possible 1. So perhaps while probability is about statistical likeliness, a more rigorous than usual view of possibility is all about contrasting absolute non-existence vs. absolute existence?
I don't really understand what it is you're after. Mathematically, probability is described by probability theory and possibility/necessity by modal logic. Those are very different things.
\[ \Diamond p \Leftrightarrow \lnot \Box \lnot p \]: "p is possible if, and only if, it is not necessary that not-p."
Possibility and necessity (let's call them M and L in text) act like quantifiers rather than values, and formally behave very analogously to existential and universal quantifiers:
\[ (\exists x)(Px) \Leftrightarrow \lnot(\forall x)(\lnot Px) \]: "there exists an x such that Px if, and only if, it is not the case that for all x, Px is false."

Originally Posted by Jorghnassen View Post
A better definition of a "possibility" would be any element (or even subset) of the probability space.
There is no reason why a probability space can't include impossible events. If I roll a standard die, there is no possibility of getting a 10. That can be put in the probability space without any trouble. And technically, the empty set is an event that is in many situations not possible.
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Last edited by Vorpal; 16th April 2012 at 12:02 PM.
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Old 16th April 2012, 12:26 PM   #42
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Originally Posted by wollery View Post
If something is possible then it has a probability greater than 0.

That's about it.
This is not quite true. It is possible to flip a fair coin an infinite number of times and have it come up heads every time. Yet the probability is zero.
http://en.wikipedia.org/wiki/Almost_surely
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Old 16th April 2012, 12:32 PM   #43
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Originally Posted by phildonnia View Post
This is not quite true. It is possible to flip a fair coin an infinite number of times and have it come up heads every time. Yet the probability is zero.
http://en.wikipedia.org/wiki/Almost_surely
How is it possible to do anything an infinite number of times?
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Old 16th April 2012, 12:41 PM   #44
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Originally Posted by phunk View Post
How is it possible to do anything an infinite number of times?
Because it's a mathematical idealization. Or something.
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Old 16th April 2012, 06:41 PM   #45
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Originally Posted by Vorpal View Post
I don't really understand what it is you're after. Mathematically, probability is described by probability theory and possibility/necessity by modal logic. Those are very different things.
\[ \Diamond p \Leftrightarrow \lnot \Box \lnot p \]: "p is possible if, and only if, it is not necessary that not-p."
Possibility and necessity (let's call them M and L in text) act like quantifiers rather than values, and formally behave very analogously to existential and universal quantifiers:
\[ (\exists x)(Px) \Leftrightarrow \lnot(\forall x)(\lnot Px) \]: "there exists an x such that Px if, and only if, it is not the case that for all x, Px is false."


There is no reason why a probability space can't include impossible events. If I roll a standard die, there is no possibility of getting a 10. That can be put in the probability space without any trouble. And technically, the empty set is an event that is in many situations not possible.
Very well explained (on both points). Thank you.
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Old 17th April 2012, 10:48 PM   #46
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Originally Posted by phildonnia View Post
This is not quite true. It is possible to flip a fair coin an infinite number of times and have it come up heads every time. Yet the probability is zero.
http://en.wikipedia.org/wiki/Almost_surely
If such thing is possible, it would be ridiculously unlikely!
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Old 18th April 2012, 12:14 AM   #47
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If you think anything's possible, try winning a game of Chess with your first move.
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Old 18th April 2012, 03:06 AM   #48
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If your opponent concedes?
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Old 18th April 2012, 07:13 AM   #49
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Originally Posted by DazzaD View Post
If your opponent concedes?
Or the clock runs out.
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Old 18th April 2012, 07:14 AM   #50
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Originally Posted by Gord_in_Toronto View Post
Or the clock runs out.
(did not even need "quantum" for that!)
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