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Old 15th April 2012, 12:32 PM   #1
eijah
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Probabilty vs. Possibility

A probability can be from 0 to 1, but what about a possibility?

That is, is a possibility either 0 or 1? Or can a possibility also be from 0 to 1? Note: I am not talking about conditional probabilities, or am I?
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Old 15th April 2012, 12:34 PM   #2
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Originally Posted by eijah View Post
Note: I am not talking about conditional probabilities, or am I?
That depends.
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Old 15th April 2012, 12:36 PM   #3
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Originally Posted by ctamblyn View Post
That depends.
Ok. Let's assume that I am not.
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Old 15th April 2012, 12:39 PM   #4
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Just joking.

Assuming this is a serious question, then, I'm not aware of a mathematical "possibility theory". The way I tend to hear it used in everyday speech, "X is possible" is basically synonymous with "the probability of X is not zero", and "X is impossible" is synonymous with "the probability of X is zero".
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Old 15th April 2012, 12:54 PM   #5
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Originally Posted by eijah View Post
can a possibility also be from 0 to 1?
In some circumstances we can reasonably indicate possibility as a percentage between 0% and 100% -- but then you can decide to interpret any percentage less than 100% as zero.

For example, if you ask whether it is possible for a factory to produce 100,000 automobiles per year, the answer might be that it is 60% possible. It can produce 60,000 per year, so it can fulfill 60% of your requested objective, which is more than 0% of the objective, after all.
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Old 15th April 2012, 12:57 PM   #6
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Originally Posted by ctamblyn View Post
Just joking.

Assuming this is a serious question, then, I'm not aware of a mathematical "possibility theory". The way I tend to hear it used in everyday speech, "X is possible" is basically synonymous with "the probability of X is not zero", and "X is impossible" is synonymous with "the probability of X is zero".
This.

Originally Posted by JJM 777 View Post
For example, if you ask whether it is possible for a factory to produce 100,000 automobiles per year, the answer might be that it is 60% possible. It can produce 60,000 per year, so it can fulfill 60% of your requested objective, which is more than 0% of the objective, after all.
No, I'd say then, it's not possible to produce the 100,000 cars.
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Old 15th April 2012, 12:59 PM   #7
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Let's take a coin flip. Let's say we have a standard American quarter and a nice solid table on which to flip it.

Common parlance is that we have two, equally probable outcomes and for even the most exact of real world scenarios this is true. If flipped into the air above the table the quarter will either land face up or face down. Since there is no force that would cause the quarter to land one side up with statistical more regularly then the other* and the factors in how the coin are going to land are far outside the ability of the person flipping it to alter, assuming we're talking a good old fashioned honest coin flip and not some kind of slight of hand, we can reasonably assume a coin flip is a fair, real world random event generator with a 50/50 split possible outcome.

But in the realm of pure hypotheticalness we have billions of possible scenarios, practically all of which are so statistically unlikely we can dismiss them when applying a coin flip to any real world scenario. The coin could land perfectly balanced on its side, it could quantum tunnel through the table and land on the floor, etc.

None of these things are likely by any possible definition of the term, so much so that even speaking of the wildest possible outcomes we can safety dismiss them as impossible with intellectual honestly.

Now let's take a bag of 100 marbles, 99 white, and 1 black that a person reaches into and sight unseen picks one. Again here we have two equally possible scenario in a practical sense, a person is going to pick either a white marble or a black one. But the probability of picking a white marble is many times greater then picking a black one.

*And yes I'm aware that due to weight issues on the engraving of some coins they will land on one side statistically more often then other by a small but not insignificant margin, but I'm trying to make a point.
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Old 15th April 2012, 01:20 PM   #8
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If something is possible then it has a probability greater than 0.

That's about it.
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Old 15th April 2012, 01:24 PM   #9
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Probability, in mathematics, is a well defined noun. With the definition comes precision concerning it properties and how it works.

Possibility is not as well defined, and particularly not well defined mathematically. Without it, it's measure is ephemeral and qualitative.
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Old 15th April 2012, 02:44 PM   #10
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On uncountable sets, events of measure 0 do occur (any realisation of a continuous random variable). So saying a "possibility" has probability bigger than 0 is technically false. A better definition of a "possibility" would be any element (or even subset) of the probability space.

But in the end, a possibility is nothing more than a colloquial term.
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Old 15th April 2012, 03:14 PM   #11
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Originally Posted by eijah View Post
Ok. Let's assume that I am not.
More cryptic you can be.
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Old 15th April 2012, 03:28 PM   #12
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Originally Posted by Jorghnassen View Post
On uncountable sets, events of measure 0 do occur (any realisation of a continuous random variable). So saying a "possibility" has probability bigger than 0 is technically false. A better definition of a "possibility" would be any element (or even subset) of the probability space.

But in the end, a possibility is nothing more than a colloquial term.
Perhaps so for some mathematical disciplines, but maybe not for all.

In any case, I don't think that saying that if something has a possibility of 1, it means that it can be or happen, whereas, if it has a possibility of 0 it it can't be or happen.

An example of what I am suggesting...

Newark Mayor Corey Booker this past week rushed into a neighbor's burning building to save her. He found himself in that burning building X with a goal of completely vacating X through an intended doorway Y to hopeful safety in the street.

As Mayor Booker and his neighbor were moving more and more towards the intended Y door to the safety of the street, the probability of their salvation was obviously increasing towards 1. But I submit to you for your response that until they wholly transitioned from being in the burning building to being in the street (where the possibility of burning in the building became 0) the possibility of them burning to death in the burning building remained 1.
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Old 15th April 2012, 04:08 PM   #13
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Originally Posted by eijah View Post
A probability can be from 0 to 1, but what about a possibility?

That is, is a possibility either 0 or 1? Or can a possibility also be from 0 to 1? Note: I am not talking about conditional probabilities, or am I?
I would say "possibility" refers to anything with a non-zero probability.

Possibility is not a statistical term, so there is no numerical measure of it (so no, a possibility cannot "also be from 0 to 1").
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Old 15th April 2012, 04:42 PM   #14
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Originally Posted by JoeTheJuggler View Post

Possibility is not a statistical term, so there is no numerical measure of it (so no, a possibility cannot "also be from 0 to 1").
I very much agree. E.g., whereas probability to me has the look and feel of a statistical concept, possibility seems to me to be much more an existential one. So how about this: An atheist would say that God has an existential possibility of 0, whereas an agnostic might say that God has a statistical probability of perhaps 0?
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Old 15th April 2012, 05:30 PM   #15
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Surely any event is possible. Some are probable; others are not.

As I was saying to the elephant in my garage.
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Old 15th April 2012, 05:42 PM   #16
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Originally Posted by Gord_in_Toronto View Post
Surely any event is possible. Some are probable; others are not.
I think there are a lot of people here in addition to me who do not think all events are possible.
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Old 15th April 2012, 05:56 PM   #17
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Originally Posted by eijah View Post
A probability can be from 0 to 1, but what about a possibility?

That is, is a possibility either 0 or 1? Or can a possibility also be from 0 to 1? Note: I am not talking about conditional probabilities, or am I?

Probability is a scalar quantity with a range from zero to one (often expressed as a percentage), while possibility is simply true/false (which can be expressed as one and zero in Boolean algebra).

A probability of zero means impossible (while a probability of one means certain). Anything not impossible is, by definition, possible. Therefore every nonzero probability is a possibility.
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Old 15th April 2012, 05:56 PM   #18
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Originally Posted by eijah View Post
I very much agree. E.g., whereas probability to me has the look and feel of a statistical concept, possibility seems to me to be much more an existential one.
That is what everyone would agree: Probability is a defined quantity in statistics. Everyone given the same data and using the same techniques will calculate the same probability (it is an objective measure).

Possibility is a English term for the subjective feeling abut how possible something is.
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Old 15th April 2012, 06:12 PM   #19
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I thought that this was resolved in Pirates of the Caribbean: The Curse of the Black Pearl.
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Old 15th April 2012, 06:40 PM   #20
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Originally Posted by Brian-M View Post
...possibility is simply true/false (which can be expressed as one and zero in Boolean algebra).
Hmm, if it were purely true/false, then it wouldn't take qualifiers, like "strongly possible" or "barely possible". There is an element of true/false about it (anything that's not impossible can be described as possible), but it's more complex than that. I wouldn't go so far as to say you're wrong, but I don't think your description tells the whole story.

Natural languages are weird. But fun.
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Old 15th April 2012, 06:57 PM   #21
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Originally Posted by xtifr View Post
Hmm, if it were purely true/false, then it wouldn't take qualifiers, like "strongly possible" or "barely possible". There is an element of true/false about it (anything that's not impossible can be described as possible), but it's more complex than that. I wouldn't go so far as to say you're wrong, but I don't think your description tells the whole story.

Natural languages are weird. But fun.
I am delighted that we have gotten this far. At least what I am referring to is not simple or for the simpleminded.

It does seem to me that if we initially dismiss fuzzy possibilities, we are left with hard possibilities of either 0 or 1. And if that is the case, how do they compare with probabilities of 0 and 1? E.g., is a possibility of 0 the same as a probability of 0? Or just equivalent? And in all domains, or just in some or in one?
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Old 15th April 2012, 07:16 PM   #22
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Originally Posted by eijah View Post
I think there are a lot of people here in addition to me who do not think all events are possible.
Think all you want. Prove it if you can.

With a bit of quantum physics and some hand waving, I can prove it to be true.
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Old 15th April 2012, 07:16 PM   #23
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Originally Posted by xtifr View Post
Hmm, if it were purely true/false, then it wouldn't take qualifiers, like "strongly possible" or "barely possible". There is an element of true/false about it (anything that's not impossible can be described as possible), but it's more complex than that. I wouldn't go so far as to say you're wrong, but I don't think your description tells the whole story.

Natural languages are weird. But fun.
When you're using qualifiers like that you're indicating probability, not possibility. "Barely possible" would indicate a probability close to (but not exactly) zero.
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Old 15th April 2012, 07:23 PM   #24
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It's perfectly explained on post #4.
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Old 15th April 2012, 07:29 PM   #25
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Originally Posted by eijah View Post
I am delighted that we have gotten this far. At least what I am referring to is not simple or for the simpleminded.

It does seem to me that if we initially dismiss fuzzy possibilities, we are left with hard possibilities of either 0 or 1. And if that is the case, how do they compare with probabilities of 0 and 1? E.g., is a possibility of 0 the same as a probability of 0? Or just equivalent? And in all domains, or just in some or in one?
"Possible" means the probability is greater than zero and less-than-or-equal-to one. At least in English. I'm not sure what other domains you think might be relevant. It's not a mathematical term; it's an English one.

Were you looking for the linguist's forum? Language Log is thataway. ->
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Old 15th April 2012, 07:43 PM   #26
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Originally Posted by Dani View Post
It's perfectly explained on post #4.
I'm afraid not. While "impossible" implies "probability of 0", the converse is not true. See any random variable with an uncountable support. There's a reason "almost sure convergence" is called "almost sure convergence".
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Old 15th April 2012, 08:42 PM   #27
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Originally Posted by Jorghnassen View Post
I'm afraid not. While "impossible" implies "probability of 0", the converse is not true. See any random variable with an uncountable support. There's a reason "almost sure convergence" is called "almost sure convergence".
I agree.

Imagine throwing an infinitely sharp dart at a dart board. The probability that it hits any particular point is zero. Nevertheless, it must hit some point.

So every point is possible, even though they all have probability zero.
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Old 15th April 2012, 09:10 PM   #28
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Originally Posted by sol invictus View Post
Imagine throwing an infinitely sharp dart at a dart board. The probability that it hits any particular point is zero.
Really? Every time I try to calculate the probability, I come up with 1/NaN, not zero. (Not that that gets us much closer to resolving the original question.)
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Old 15th April 2012, 09:28 PM   #29
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Originally Posted by Jorghnassen View Post
I'm afraid not. While "impossible" implies "probability of 0", the converse is not true. See any random variable with an uncountable support. There's a reason "almost sure convergence" is called "almost sure convergence".
Why not? As far as I understand the law of great numbers, I still think it's a valid and very useful semantic equivalence. After all, "possible" doesn't mean that it's going to happen.
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Old 15th April 2012, 09:28 PM   #30
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Originally Posted by sol invictus View Post
I agree.

Imagine throwing an infinitely sharp dart at a dart board. The probability that it hits any particular point is zero. Nevertheless, it must hit some point.

So every point is possible, even though they all have probability zero.
You're assuming that the points on the dart board are infinite in number (and infinitely small), instead of being a finite number of different-colored contiguous regions.



ETA: Yes, I know that a point is infinitely small by definition. But these theoretical points don't exist physically, while dartboards do.
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Old 15th April 2012, 09:37 PM   #31
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Originally Posted by sol invictus View Post
I agree.

Imagine throwing an infinitely sharp dart at a dart board. The probability that it hits any particular point is zero. Nevertheless, it must hit some point.

So every point is possible, even though they all have probability zero.
Ok, I'm no statistician, and I'm lost here. Probability zero? Why? I would say the probability is very close to zero, which I would translate into "almost impossible".

Slightly off topic question:

Can someone explain with examples the difference between the weak and strong law of great numbers? I've been unable to grasp it.
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Old 15th April 2012, 09:48 PM   #32
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Originally Posted by xtifr View Post
Really? Every time I try to calculate the probability, I come up with 1/NaN, not zero.
Yes, really: you have a probability measure μ that assigns sets of points the probability of the dart hitting a point in that set. For a uniform probability, the probability measure of a set will be proportional to its area. Then it's direct that for any individual point p, μ({p}) = 0.

In general, it's possible for the probability of individual points to be nonzero, but there can't be uncountably many such points, or else the probability of the whole space would not be 1. But those kinds of things are not useful for modeling realistic dartboards.
And God help you if your probability measure is not σ-additive or has non-measurable singletons or something.
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Old 15th April 2012, 09:49 PM   #33
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Originally Posted by Dani View Post
Ok, I'm no statistician, and I'm lost here. Probability zero? Why?
Because the dart would have to be infinitely narrow in order to be infinitely sharp, meaning it could only hit a single point. But points are infinitely small, meaning that there would be infinite points on the dartboard. So your chances of hitting a specific point would be one in infinity, or zero.

You could make the argument that it's almost zero but isn't actually zero, but that's a topic for another thread. Like this one, which debates whether or not 0.9999... (with infinite nines) is the same thing as one.
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Old 15th April 2012, 10:30 PM   #34
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Originally Posted by Brian-M View Post
Because the dart would have to be infinitely narrow in order to be infinitely sharp, meaning it could only hit a single point. But points are infinitely small, meaning that there would be infinite points on the dartboard. So your chances of hitting a specific point would be one in infinity, or zero.

You could make the argument that it's almost zero but isn't actually zero, but that's a topic for another thread. Like this one, which debates whether or not 0.9999... (with infinite nines) is the same thing as one.
Thanks for the explanation. Yes, but that's just theoretical. The fact though is that... ok... I just read what you added in your response to Sol Invictus, and that's what I was going to say.

For all purposes, and with some nuances if you wish (basically, that probability is scientific and possibility is colloquial) the question has been satisfactorily answered.
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Old 15th April 2012, 10:42 PM   #35
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Originally Posted by eijah View Post
1) A probability can be from 0 to 1, but what about a possibility?

2) That is, is a possibility either 0 or 1? Or can a possibility also be from 0 to 1? Note: I am not talking about conditional probabilities, or am I?
Probability basically tells you the likelihood of an event. For instance, A probability of let's say 0.5 means that the event will occur 50% of the time.

2) Probability and possibility are two different things. Probability tells you the likelihood of an event while possibility means well...possible
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Old 16th April 2012, 04:52 AM   #36
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Originally Posted by Dani View Post
Why not? As far as I understand the law of great numbers, I still think it's a valid and very useful semantic equivalence. After all, "possible" doesn't mean that it's going to happen.
Well, if you call it the law of "great" numbers, you either don't know it or studied it in another language than English (where it is known as the law of large numbers). For any continuous random variable, i.e. any quantity you can measure with indefinite precision without counting, you cannot put any positive weight to a single point, because the axioms of probability need the sum of those weights to be 1, and you can only do that on quantities that are countable. For example, the probability that it is 15 degrees outside is zero, because it could be 15.01, 14.99, 15.001, 15.0001, 14.999, etc. ad infinitum if we could measure it with enough precision (and it is impossible to enumerate all the possible values of the temperature). So the probability of any point is actually 0.
So the temperature of 15 degrees is possible, but it has probability 0. However, you can give a positive probability to an interval, i.e. the set of all points between a and b (say), and you can define with that a probability density function, which has similar properties to probability mass functions, but assigns a density, not probability, to a point.
So, while there is a temperature outside, and it has to take a value, the probability of any value for that temperature is 0, though the probability that it falls between some values a and b can be positive. However, some temperatures are impossible, particularly anything below absolute zero. Those temperatures have probability 0 and density 0 as they fall outside the support of the temperature range.

Of course, continuous random variables are not the only "creatures" where possible and probability 0 intersect. They are just the most common (and realistic) examples.
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Old 16th April 2012, 06:34 AM   #37
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Jorghnassen:

No, I haven't studied the law of large numbers, I'm just familiar with it. However my confusion was due to the fact that I'm Spanish and didn't remember that in English it's large instead of great. I urge people to correct me in my signature, so thanks.

And thanks for the explanation too. It helped me, and you're right. In fact, while I was going to work (hey, I'm Spanish and I have a job! Talk about probabilities... ) I was thinking of this and getting to your and Sol Invictus' conclusion through a similar - although less elaborated - reasoning.
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Old 16th April 2012, 07:55 AM   #38
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Originally Posted by Jorghnassen View Post
Well, if you call it the law of "great" numbers, you either don't know it or studied it in another language than English (where it is known as the law of large numbers). For any continuous random variable, i.e. any quantity you can measure with indefinite precision without counting, you cannot put any positive weight to a single point, because the axioms of probability need the sum of those weights to be 1, and you can only do that on quantities that are countable. For example, the probability that it is 15 degrees outside is zero, because it could be 15.01, 14.99, 15.001, 15.0001, 14.999, etc. ad infinitum if we could measure it with enough precision (and it is impossible to enumerate all the possible values of the temperature). So the probability of any point is actually 0.
So the temperature of 15 degrees is possible, but it has probability 0. However, you can give a positive probability to an interval, i.e. the set of all points between a and b (say), and you can define with that a probability density function, which has similar properties to probability mass functions, but assigns a density, not probability, to a point.
So, while there is a temperature outside, and it has to take a value, the probability of any value for that temperature is 0, though the probability that it falls between some values a and b can be positive. However, some temperatures are impossible, particularly anything below absolute zero. Those temperatures have probability 0 and density 0 as they fall outside the support of the temperature range.

Of course, continuous random variables are not the only "creatures" where possible and probability 0 intersect. They are just the most common (and realistic) examples.
Very helpful.

But perhaps also helpful to look at the question of probability vs. possibility in this different way...

Let's say that you have a roulette wheel with 36 slots, labeled 1 through 36. With one atypical exception. And the exception is that there is no label on the 35th slot. The probability of landing on one of the slots is of course 1/36. And the probability of landing on slot 1 is 1/36. And the probability of landing on the 35th slot is 1/36.

OTOH, the possibility of landing on a slot labeled 35 is 0.

Whereas, the possibility (NOT probability) of landing on a slot labeled 3 is 1. As is the possibility of landing on a slot labeled 3 or a slot labeled 4.

And if so, perhaps if two of the slots were labeled 6, then the possibility (NOT probability) of landing on a a slot labeled 6 is also 1.

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?
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Old 16th April 2012, 11:32 AM   #39
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Originally Posted by Jorghnassen View Post
On uncountable sets, events of measure 0 do occur (any realisation of a continuous random variable). So saying a "possibility" has probability bigger than 0 is technically false. A better definition of a "possibility" would be any element (or even subset) of the probability space.

But in the end, a possibility is nothing more than a colloquial term.
Good point, I was sloppy
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Old 16th April 2012, 11:48 AM   #40
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I have a feeling that this line of questioning is from the usage of the two terms in society.

If I were to say that there is a probability of something happening, the idea is that it will. If I say that there is a possibility that something will happen, then generally it won't.

These aren't the true definitions of the word, but just the feeling that they give.
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