Is Zero Probability Possible?

[lax1113]

I hope this belongs here...
I believe that I read in some book that there is no such thing as zero probablity events. The example I remember was that if you take the pieces of a broken vase and throw them on the ground, their is a chance that it will put itself back together. The chance is astronomically small, but there none the less. Which gives way to the idea that things peopel perceive as impossible actually aren't. I believe the other one was running through a wall by the electrons of you and the wall lining up perfectly. Am i crazy for stating this? Or did i misread/completely make something up... Sorry I can't remember the book or article, but I do believe it had something to do with quantum mechanics.

 

[137]

zero probability is taught in schools, just as imaginary numbers are.
http://www.ics.uci.edu/~liang/courses/s67-spr06/lectures/week1.pdf (last page)

its reasonable to assume that it is impossible to have a zero probability event. you throw the vase, and through some miracle of nature, god, science, each piece lands together like a puzzle. superman krazy glues it mid air (not changing the trajectory of coarse) wait...

its reasonable to assume that it is impossible to have a zero probability event.

impossible = zero probability

you may assume it is impossible to have an impossible event. english teaches that a double negative makes a positive. so it is also possible to have a possible event. this is the point where my brain goes on strike. is this misunderstood in science or a possibility our language can't express?

 

[ice109]

yea the basic assumption of statistical mechanics is that "given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates."[wiki]

a microstate is a configuration of the system.

but the probability of a system being in such a state that you can pass through a wall is astronomical, it's on the order of $$1/10^{23^{23}}$$

and so even if you check the system that is you and the wall every second since the beginning of time, or even every Planck second you'd have to wait till after the universe dies a cold cold death before you even have the chance of seeing that happen. and that's just for the chance of it happening. maybe this go around wasn't your lucky Planck second and you have to wait another infimum of of Planck seconds until it happens.

yea it's impossible. when it's that close to zero chance that it'll happen you might as well say there's zero chance of it happening.

 

[lax1113]

I understand what you mean about the wall thing, the chances are so insanely small that considering it impossible is not wrong. However, I am understanding correct that technically, it isn't?

 

[Nick89]

In the case of the wall I think you might be referring to quantum tunneling, an effect where a particle can travel through a potential barrier (such as a 'wall' you might say) even though it does not have enough energy.
You can think of it like a (motorless) car being pushed up a hill. In a classical sense, if it does not have enough energy (speed) to make it over the hill, it won't make it over the hill, but it will stop somewhere and return backwards.
In quantum mechanics, if you replace the car with a particle and the hill with a certain potential 'hill' (such as an electrostatic force or something) there is indeed a chance that the particle makes it over the hill even if it did not have enough energy.

I believe that it is possible for you to tunnel through a wall. However, the chance that just ONE particle would tunnel through the wall (which is insanely wide if you look at it from a single particles point of view) is already astronomically small, and if you want to tunnel through yourself, every single particle in your body would have to tunnel at the exact same time. The chances of that happening are so small that you can safely ignore it in pretty much every application. (The chance however is not zero, so if you were to walk at a wall right now you could in fact tunnel through! That would be quite an achievement indeed... Good luck ;) )I don't know, but I think there are certain things that you can say have exactly zero probability. Imagine (an often seen example) a vase with a number of balls. Now imagine that every ball is red. What is the chance of you drawing a yellow ball out of the vase? As far as I know there is nothing that can make the atoms in the balls change spontaneously so they suddenly reflect only yellow wavelengths instead of only red wavelengths, so I think you can safely say that chance is zero.

 

[ice109]

I understand what you mean about the wall thing, the chances are so insanely small that considering it impossible is not wrong. However, I am understanding correct that technically, it isn't?

technically doesn't mean anything.

In the case of the wall I think you might be referring to quantum tunneling, an effect where a particle can travel through a potential barrier (such as a 'wall' you might say) even though it does not have enough energy.
You can think of it like a (motorless) car being pushed up a hill. In a classical sense, if it does not have enough energy (speed) to make it over the hill, it won't make it over the hill, but it will stop somewhere and return backwards.
In quantum mechanics, if you replace the car with a particle and the hill with a certain potential 'hill' (such as an electrostatic force or something) there is indeed a chance that the particle makes it over the hill even if it did not have enough energy.

I believe that it is possible for you to tunnel through a wall. However, the chance that just ONE particle would tunnel through the wall (which is insanely wide if you look at it from a single particles point of view) is already astronomically small, and if you want to tunnel through yourself, every single particle in your body would have to tunnel at the exact same time. The chances of that happening are so small that you can safely ignore it in pretty much every application. (The chance however is not zero, so if you were to walk at a wall right now you could in fact tunnel through! That would be quite an achievement indeed... Good luck ;) )

yea that's another instance where the probability exponentially decays as the wall gets thicker.

I don't know, but I think there are certain things that you can say have exactly zero probability. Imagine (an often seen example) a vase with a number of balls. Now imagine that every ball is red. What is the chance of you drawing a yellow ball out of the vase? As far as I know there is nothing that can make the atoms in the balls change spontaneously so they suddenly reflect only yellow wavelengths instead of only red wavelengths, so I think you can safely say that chance is zero.

umm there's no chance of that period,

 

[ThomasT]

I hope this belongs here...
I believe that I read in some book that there is no such thing as zero probablity events. The example I remember was that if you take the pieces of a broken vase and throw them on the ground, their is a chance that it will put itself back together. The chance is astronomically small, but there none the less. Which gives way to the idea that things peopel perceive as impossible actually aren't. I believe the other one was running through a wall by the electrons of you and the wall lining up perfectly. Am i crazy for stating this? Or did i misread/completely make something up... Sorry I can't remember the book or article, but I do believe it had something to do with quantum mechanics.

The sorts of possibilities you're talking about are only possibilities in an imaginary space used to model microstates in statistical mechanics.

Whether or not it really is possible for a broken vase to spontaneously reassemble itself is open to speculation. But the odds are not good that this is a real possibility in the real world.

As for quantum weirdness, it's a function of how the theory might be interpreted or how the experimental results might be translated into the ordinary language of our sensory experience. The fact is that nobody knows if there's anything weird about the processes underlying experimental results. There are reasons to believe that the wave mechanics of some underlying quantum realm are essentially the same as the macroscopic or classical wave mechanics of our sensory experience.

 

[Manchot]

The premise of this thread is flawed. There are definitely situations in which you will find probabilities to be exactly zero in QM (e.g., the probability of finding the first excited state of a square well in the center of the well).

 

[dkgolfer16]

I hope this belongs here...
I believe that I read in some book that there is no such thing as zero probablity events.

I have a baseball. I prepare to throw it. There is zero probablity that I throw it faster than the speed of light. Zero probablity event.

 

[pallidin]

I have a baseball. I prepare to throw it. There is zero probablity that I throw it faster than the speed of light. Zero probablity event.

Why? Maybe you DID throw it FTL, but now it's back in your hands before you threw it.

 

[DaveC426913]

technically doesn't mean anything.

The appropriate phrase is 'in principle'. As in:

While the chances of passing through a wall are astronomically small, it is possible in principle.

Why? Maybe you DID throw it FTL, but now it's back in your hands before you threw it.

The point is that the universe, as we currently understand it, physically forbids this event. It cannot happen.

I don't know, but I think there are certain things that you can say have exactly zero probability. Imagine (an often seen example) a vase with a number of balls. Now imagine that every ball is red. What is the chance of you drawing a yellow ball out of the vase? As far as I know there is nothing that can make the atoms in the balls change spontaneously so they suddenly reflect only yellow wavelengths instead of only red wavelengths, so I think you can safely say that chance is zero.

Spontaneous decay of protons in the atoms of the dyes that the balls are coloured with.

It is about as likely as passing through a wall.

 

[dkgolfer16]

Why? Maybe you DID throw it FTL, but now it's back in your hands before you threw it.

Your saying I did throw it and I didn't throw it. I don't understand. And since when can a baseball travel FTL? Do you have experimental proof?

 

[pallidin]

Yeah, good point Dave.
Dkgolfer16... see Dave's reply in post #11

 

[dkgolfer16]

Yeah, good point Dave.
Dkgolfer16... see Dave's reply in post #11

Let's get back to the basics. What was the meaning of your original response to my first post? It sounded like you were saying there is a chance a baseball can travel FTL.

 

[pallidin]

dk, Dave corrected me which is why I referenced his post to you.

 

[pallidin]

Even still, "zero-probability" is a brazen concept, as it implies immutable understanding and certainty for all potential aspects of an event which has not yet occurred.

 

[DaveC426913]

Even still, "zero-probability" is a brazen concept, as it implies immutable understanding and certainty for all potential aspects of an event which has not yet occurred.

I do think the baseball is a good example.

 

[andrewm]

Even still, "zero-probability" is a brazen concept, as it implies immutable understanding and certainty for all potential aspects of an event which has not yet occurred.

Physics has never perfectly described the physical world. It is a science of approximations. (It is only approximations!). In physics, there are many P=0 events. And in the real world? I don't know, but that sort of question isn't physics at all! It's philosophy.

 

[dkgolfer16]

I do think the baseball is a good example.

Thanks Dave.

 

[Seventy-Eight]

Events of probability zero are a rather subtle concept, and as a result having probability zero is not exactly the same as impossible. For example, for a continuous random variable, the probability of taking any particular value is zero. The probability of observing a path of a Brownian motion that can be differentiated is zero. Probability theory is just a particular application of measure theory, and a probability of event zero is the same as a volume of zero. Point particles have a volume of zero, and they are quite interesting things.

Also, the question assumes that events have a physical probability that can somehow be measured. Probability is not a physical property, but instead reflects our knowledge of an event. From that point of view we can take probability 1 events to be those that we know are unambiguously true, and probability 0 events those that we know are untrue. So for example I know my age with probability one, but for you it has some probability of being anything from 0 up.

 

[silkop]

Events of probability zero are a rather subtle concept, and as a result having probability zero is not exactly the same as impossible. For example, for a continuous random variable, the probability of taking any particular value is zero.

Only if you use sloppy language. The non-sloppy and non-confusing statement is that probability tends towards zero in the limit as you reduce the interval's size towards 0. Furthermore, it seems there are no continuous variables in the real world; for all practical purposes actual phenomena are discrete, and assuming variables to be continuous is just a mathematical convenience.

Also, the question assumes that events have a physical probability that can somehow be measured. Probability is not a physical property, but instead reflects our knowledge of an event.

Correct.

From that point of view we can take probability 1 events to be those that we know are unambiguously true, and probability 0 events those that we know are untrue. So for example I know my age with probability one, but for you it has some probability of being anything from 0 up.

Riight, though in science we are mostly concerned with probabilities conditioned on the total information of everyone, including assumptions shared by everyone based on evidence which has accumulated so far over humanity's entire history. For better or worse, this mode of thinking is implicit when we attempt to attribute probability to some real-world occurrences.

 

[Hurkyl]

Only if you use sloppy language. The non-sloppy and non-confusing statement is that probability tends towards zero in the limit as you reduce the interval's size towards 0.

The only sloppiness in Seventy-eight's language is talking about a random variable 'taking a value' (which is a normal abuse of language). He is perfectly correct in stating

P(X=x) = 0.​

There are no hidden limits involved in that statement -- it is simply applying the probability measure to the one-point event {x}.

Furthermore, it seems there are no continuous variables in the real world; for all practical purposes actual phenomena are discrete, and assuming variables to be continuous is just a mathematical convenience.

There is no evidence for such an assertion. If 'actual phenomena are discrete', then there would be side effects. No such side effects have been observed to date. The experimental evidence tells us either that 'actual phenomena' can be continuous, or at least that reality is many orders of magnitude more continuous than a "for all practical purposes" discretization.

Riight, though in science we are mostly concerned with probabilities conditioned on the total information of everyone, including assumptions shared by everyone based on evidence which has accumulated so far over humanity's entire history.

No we're not. If that were true, it would be mostly impossible to practice science, because such a level of information is unwieldy and unobtainable.

 

[silkop]

The only sloppiness in Seventy-eight's language is talking about a random variable 'taking a value' (which is a normal abuse of language). He is perfectly correct in stating

P(X=x) = 0.​

There are no hidden limits involved in that statement -- it is simply applying the probability measure to the one-point event {x}.

When you talk about a measure on an infinite set, you have already made the "hidden limit" step without noticing. There is no empirical evidence for the real existence of anything like "infinite set", nor is it conceivable what sort of evidence would be required. Infinite sets, and continuous variables likewise, can and should be viewed as purely hypothetical outcomes of generating processes (whose specification is finite, which explains why our finite brains can deal with them quite well).

There is no evidence for such an assertion. If 'actual phenomena are discrete', then there would be side effects.

Please name a few. Also explain why you postulate observable side effects given imperfect instruments used for observations.

No such side effects have been observed to date.

Very much has not been observed to date at the microscopic (nanoscopic?) level.

The experimental evidence tells us either that 'actual phenomena' can be continuous, or at least that reality is many orders of magnitude more continuous than a "for all practical purposes" discretization.

I agree with you about the high granularity. However, if I'm not mistaken, the more granular you get, the more discrete things appear to be. By "all practical purposes" I meant areas like engineering and computing applications, where infinite concepts out of necessity are shrugged off in favor of good discrete approximations.

No we're not [concerned with probabilities conditioned on all available information while doing science]. If that were true, it would be mostly impossible to practice science, because such a level of information is unwieldy and unobtainable.

My statement was meant as an idealization - another "passing into the limit", if you will. How else do you explain the holy grail of "objectiveness" pursued by science? Another way of stating it is that as scientists we strive not to leave out available relevant information from our inferences. This approach does not need to become unwieldy for two reasons 1) newer refined models always include older less precise models as a special case; their increment in information is modest 2) some information "cancels out" - a highly detailed model may be no better in making predictions than a simple model. In fact, parsimonious models are very much preferable given our limited, finite resources.

 

[Hurkyl]

When you talk about a measure on an infinite set, you have already made the "hidden limit" step without noticing. There is no empirical evidence for the real existence of anything like "infinite set", nor is it conceivable what sort of evidence would be required. Infinite sets, and continuous variables likewise, can and should be viewed as purely hypothetical outcomes of generating processes (whose specification is finite, which explains why our finite brains can deal with them quite well).

The statement you're replying to has absolutely nothing to do with things like "empirical evidence" or "real existence" or anything like that. It's merely a mathematical statement.

Besides, I haven't a clue how, if any of this actually were relevant, would support your claims.

Please name a few.

The one I recall is that accumulated errors over the eons would make distant objects appear blurry. However, distant objects do not appear blurry.

Also explain why you postulate observable side effects given imperfect instruments used for observations.

I don't postulate anything. I am repeating (what I recall of) the writings of scientists who have done research on the topic.

What point is there in mentioning "imperfect instruments"? Statisticians have known for ages how to deal with noisy observations.

Very much has not been observed to date at the microscopic (nanoscopic?) level.

I don't understand this. It's almost like you're trying to infer
. The universe is discrete
from the hypotheses
. We lack evidence of the universe being discrete
. We lack evidence of other things too
but that's obviously a nonsensical argument...

I agree with you about the high granularity. However, if I'm not mistaken, the more granular you get, the more discrete things appear to be.

In what way?