Probabilty vs. Possibility

[eijah]

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?

 

[Gord_in_Toronto]

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.

 

[Brian-M]

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.

 

[Dani]

It's perfectly explained on post #4.

 

[xtifr]

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. ->

 

[Jorghnassen]

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".

 

[sol invictus]

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.

 

[xtifr]

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.)

 

[Dani]

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.

 

[Brian-M]

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.

 

[Dani]

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.

 

[Vorpal]

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.

 

[Brian-M]

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.

 

[Dani]

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.

 

[Musibrique]

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

 

[Jorghnassen]

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.

 

[Dani]

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.

 

[eijah]

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?

 

[ctamblyn]

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

 

[Careyp74]

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.