Does a Probability of 1 Really mean a Dead Cert

[bhobba]

Hi Guys

I normally post over on the Quantum Physics subforum and over there the question came up of if a probability of 1 means a dead cert. I am pretty sure Kolmogrov's axioms imply it but another person wasn't so sure. Here is what he said:

'I think it will be easier to explain this by example. Consider random process that produces series of red points somewhere in a unit disk with uniform probability density. The probability of the event that the next point will concide with any point A of the disk is equal to 0.

However, after the event occurs, some point of the disk will be red. At that instant, an event with probability 0 has happened.

Actually, all events that happen in such random process are events that have probability 0.

So "event has probability 0" does not mean "impossible event".

Similarly, "probability 1" does not mean "certain event". Consider probability that the red point will land at point with both coordinates irrational. This can be shown to be equal to 1 in standard measure theory. However, there is still infinity of points that have rational coordinates, and these can happen - they are part of the disk.

In the language of abstract theory, all this is just a manifestation of the fact that equal measures do not imply that the sets are equal.'

My view is since physically you can't tell an irrational from a rational by measurement (that would involve infinite precision) and since the rationals have Lebesgue measure zero it's not a well defined question - at least physically.

Anyway - what do people think - does probability 1 imply a dead cert.

Oh - and this came up in relation to determinism being contained in a probabilistic theory - but it only has probabilities of zero and 1. BTW if that isn't true then the Kochen-Specker theorem is in big trouble - because that's an assumption it makes.

Thanks
Bill

 

[pwsnafu]

You need to be able to distinguish between the physical reality itself and the mathematical model of it.

Example: consider a coin toss. Assume we really do have a fair coin (if not we can just toss twice) in our hand. Now the mathematics says "the probability of a head or tail is 1" and it's a discrete probability space. So from the model's perspective, sure, it's a certainty. But in reality (no pun intended) there is always the extremely small chance of the coin landing on its side. The mathematical model doesn't encapsulate everything about the reality (no model can); even if it is prob 1 that doesn't mean dead certain from the perspective of reality.

I bring this up because you wrote about measurement from a physical standpoint. Dead certain is statement only about the model not reality itself. You can't dismiss an argument with "that's irrelevant in reality" when your conclusion is dependent on it.

 

[Stephen Tashi]

the question came up of if a probability of 1 means a dead cert.

What do you mean by a dead certainty? Can you express this idea without using the concept of probability? Probability theory says nothing specific about whether an event will or will not happen. The theorems of probability theory give results about the probability of events, not about the defnite truth or falsity of whether they happen.

From an applied math point of view, when a future event has probability 1, it's common to regard it as something that must occur. Probability theory (expressed as measure theory) makes no comment on the matter.

To summarize the other guys argument in a simpler situation:
Let X be a random variable that has a uniform distribution on [0,1] Each realization of X is an event with probability 0. According to measure theory, the measure of the event "The realization of X is an irrational number" is an event with probability 1. However, we cannot assume that a realization of X will not be a rational number just because this event has probability 0 since any specific realization of X has probability 0. So a probability of 0 does not preclude the possibility of an event.

That argument is introducing notions like "possibility" that are not part of probability theory. (It wouldn't be surprising if some mathematicians have invented a theory that adds "possibility" to the picture, but such math isn't part of standard measure theory.)

My view is since physically you can't tell an irrational from a rational by measurement (that would involve infinite precision) ll

That limitation neither confirms or refutes what the other guy said, but I agree with your statement. Relative to reality, the idea of a random sample is an idealization, just like the idea of a mathematical "point". We can't actually take a random sample from a continuous random variable such as the uniform distribution on [0,1]. To take such a sample, we'd have to observe a point in [0,1] with infinite precison. We can't empirically confirm or refute a result in probablity theory that assumes we take such samples.

The fact that we cannot locate points with infinite precision doesn't prevent geometry from being of practical use. I suppose the general idea of applied math is that as we approximate reality more and more precisely, our results should approach those predicted by a good mathematical theory that assumes we operate with infinite precision. Some mathematics can be tested emprically in such a manner.

However, if you look at the problem of deciding whether a random sample from a uniform distribution on [0,1] is a rational or irrational number, the idea of approximating this by using more and more precison doesn't make sense - at least by any typical definition of approximation. Every irrational point has rational points arbitrarily close to it and vice versa. Perhaps some clever mathematician of the future will invent a new definition for approximation that could be used in such situations.

 

[bhobba]

What do you mean by a dead certainty? Can you express this idea without using the concept of probability? Probability theory says nothing specific about whether an event will or will not happen. The theorems of probability theory give results about the probability of events, not about the defnite truth or falsity of whether they happen.

From the Kolmogrov axioms is the assumption of unit measure for the entire sample space ie the probability that some elementary event in the entire sample space will occur is 1. If a particular event has a probability of 1 then in any trial, or whatever situation you are modeling if the word 'trial' is not applicable, then that event must occur. One could view the sample space as containing just one event and it has probability 1. So I would say a dead cert is a situation that can be modeled with a sample space of just that outcome.

From an applied math point of view, when a future event has probability 1, it's common to regard it as something that must occur. Probability theory (expressed as measure theory) makes no comment on the matter.

I think you hit the nail on the head. Applying probability theory to QM the usual applied math point of view is assumed. In fact the usual pictorial view of the situation is the so called ensemble interpretation. Here it is assumed you have some very large ensemble of possible outcomes and the probability of an outcome occurring is the proportion of those outcomes in the ensemble. When an observation is made you randomly select one of the outcomes. If it is probability 1 then you only have that outcome in the ensemble so that must be the one selected.

Thanks
Bill

 

[pbuk]

I think it will be easier to explain this by example. Consider random process that produces series of red points somewhere in a unit disk with uniform probability density. The probability of the event that the next point will concide with any point A of the disk is equal to 0.

However, after the event occurs, some point of the disk will be red. At that instant, an event with probability 0 has happened.

This is not correct. A point is a mathematical concept with zero dimensions and cannot have any physical properties such as 'redness'. Also if some part of the disk is red, then that redness cannot be identified to a point firstly because the the location of the redness spans an infinite number of points and secondly because the location of the redness cannot be determined precisely due to Heisenberg's Uncertainty Principle.

No matter how hard you try you cannot come up with a physical event that has a probability of zero. It is also true that you cannot come up with a physical event that has a probability of 1, so there are no 'dead certs' in the real world.

 

[pbuk]

That was a bit unclear, of course it is possible to come up with a physical event that has a probability of zero - for instance the probability that two objects which are two light years apart now will be one light year apart tomorrow. What I meant was that you cannot come up with a physical event that is possible that has a probability of zero.

 

[Jano L.]

We cannot neglect a set just because it has zero measure. Measure is just one characteristic of a set, not always defined and not so important to state the fact that the set is not empty.

What I was trying to say is this: there is a fundamental difference between statement "position is x" and "position is x with probability 1" in a theoretical sense, and the former does not follow from the latter.

This is just a statement about other statements. Observations in physics have nothing to do with it.

 

[D H]

Example: consider a coin toss. Assume we really do have a fair coin (if not we can just toss twice) in our hand. Now the mathematics says "the probability of a head or tail is 1" and it's a discrete probability space. So from the model's perspective, sure, it's a certainty. But in reality (no pun intended) there is always the extremely small chance of the coin landing on its side.

That's not a good example. Probability theory man dates that P(Ω)=1, that all possible outcomes are described. Heads and tails alone do not describe the universe of possibilities of a coin toss. Per Murray and Teare (Murray and Teare, The probability of a tossed coin falling on its edge, Phys. Rev. E. 2547-2552 (1993)), "the probability of an American nickel landing on edge is approximately 1 in 6000 tosses".

BTW if that isn't true then the Kochen-Specker theorem is in big trouble - because that's an assumption it makes.

The probability of drawing exactly any specific number between 0 and 1 (inclusive) from U[0,1] is 0. Suppose you happen to draw 1/2 from that distribution. Even though the a priori probability of that event was zero, that event did happen. Compare with the probability of drawing 42 from U[0,1]. That's a truly impossible event. Drawing 1/2 from U[0,1] almost never happens, but drawing 42 from U[0,1] never happens.

This is a non-issue with finite-sized sample spaces. You only run into trouble with the distinction between almost never and never when the sample space contains an infinite number of outcomes. From what I read, the Kochen-Specker theorem appears to make an assumption of 0 and 1 as being the only possible outcomes. Correct me if I'm wrong. If that's the case, your concern over almost never versus never doesn't apply.

 

[lavinia]

Technically a zero probablitity event or collection of events has measure zero. One way to interpret this in practice is to take independent samples from the probability space and compute the average number of times the sample comes from the set of measure zero. for large samples this average will converge to zero.

So even though it is possible for the event to occur, on average it never happens.

 

[Stephen Tashi]

for large samples this average will converge to zero.

Perhaps from a physicist's point of view, it "will" converge to zero, but I think the mathematical theorem only says that it converges to zero "with probability 1", so the dragon rears its head again.

Jano L is correct (this being the mathematics section); the statement "The probability of statement A being true is 1" does not allow one to infer that "Statement A is true" if we are using the usual axioms for probability theory. Nor does the statement "Statement A is true with probabilty 0" allow us to infer "Statement A is false". ( People have 'probably' invented logics and axioms that attempt to sort out the relation between the probability of a statement and its definite truth or falsity and make explicit the distinction that D H mentions between events that are impossible and events that have zero probability. I'm only saying that the standard approach to probability theory does not deal with such things.)

 

[pbuk]

The problem is in the context - Jano L is extending statements inferred in axiomatic probability theory to the physical world by saying that there are real-world events with zero probability that actually happen, and trying to give an example of one.

A consistent statement within an axiomatic framework is not the same thing as a truth in the real world, no matter how well the maths appears to model the real world.

 

[Stephen Tashi]

The problem is in the context - Jano L is extending statements inferred in axiomatic probability theory to the physical world by saying that there are real-world events with zero probability that actually happen, and trying to give an example of one.

.

I have the impression that this thread is a spill-over from some thread in a physics section of the forum. As far as a problem "in the context", the context of the current thread (seeing that it is in the mathematics section) is mathematics - axioms, proofs and stuff like that. So I see no problem with Jano L's post in this thread.

 

[Dmobb Jr.]

If you flip a coin an infinite amount of times, the probability that you will flip heads every time is 0. It is certainly a possible outcome, though.

 

[ModusPwnd]

If you flip a coin an infinite amount of times, the probability that you will flip heads every time is 0. It is certainly a possible outcome, though.

How do you figure? I would think that if you got zero every time then you haven't actually flipped it an infinite number of times.

 

[pbuk]

If you flip a coin an infinite amount of times, the probability that you will flip heads every time is 0. It is certainly a possible outcome, though.

It's not a possible outcome because it's not a possible trial - you cannot flip a coin an infinite number of times. I assert that a trial with an outcome with a non-zero, infinitesimal probability cannot exist in the real world.

 

[Dmobb Jr.]

It's not a possible outcome because it's not a possible trial - you cannot flip a coin an infinite number of times. I assert that a trial with an outcome with a non-zero, infinitesimal probability cannot exist in the real world.

Whether something actually axists in "real world" is kind of irrelivant. "Probability 1" is a concept that we made up. It does not exist in the real world, it is simply a way of modeling it.

As far as I know everything in the world we live in is quantized. Therefore for "real world" situations it is true that probability 1 implies certainty. I think that in most situations, however, it is unreasonable to consider the world as quantized.

 

[D H]

It's not a possible outcome because it's not a possible trial - you cannot flip a coin an infinite number of times. I assert that a trial with an outcome with a non-zero, infinitesimal probability cannot exist in the real world.

This thread is in the mathematics rather than physics section of PhysicsForums. While the sciences use mathematics to describe reality, mathematics itself is not science. Mathematicians (and mathematics) don't care about the "real world".

 

[Dmobb Jr.]

This thread is in the mathematics rather than physics section of PhysicsForums. While the sciences use mathematics to describe reality, mathematics itself is not science. Mathematicians (and mathematics) don't care about the "real world".

While I basically agree with this and think that we should be considering more than just "real world", it comes to mind that we must at least consider the real world because I don't see how you could define "certainty" mathematically.

 

[Dmobb Jr.]

This article has no proofs or sources. That being said this guy sounds like he knows what he is talking about. It says in this article that you can not have an infinite set with each element having equal probability.

http://www.askamathematician.com/20...ch-one-has-an-equal-chance-of-being-selected/

Dumb post, I didn't take the 2 seconds to reallize the difference between integers and reals.

 

[D H]

I don't see how you could define "certainty" mathematically.

That is almost certainly an event with a probability of 1.

Axiomatic probability theory is based on measure theory. Measure theory helps answer questions such as what is the integral from 0 to 1 of the function f(x) whose value is 1 if x is rational, 0 if it is irrational. This leads to some interesting features such as a non-null set that nonetheless has zero measure.

 

[Dmobb Jr.]

That is almost certainly an event with a probability of 1.

What is?

 

[D H]

This article has no proofs or sources. That being said this guy sounds like he knows what he is talking about. It says in this article that you can not have an infinite set with each element having equal probability.

http://www.askamathematician.com/20...ch-one-has-an-equal-chance-of-being-selected/

You misread. There is nothing wrong with U[0,1]. The article explicitly mentioned that there is nothing wrong with this distribution. What you cannot do is construct a probability distribution on the integers such that each integer has an equal probability of being selected.

 

[pbuk]

The article also points out that while it is possible to define the uniform probability distribution over [0,1] (actually it talks about (0,1), but I'll let that ride), any attempt to sample without bias (in finite time) from that set is "doomed to failure".

 

[Mandelbroth]

This thread is in the mathematics rather than physics section of PhysicsForums. While the sciences use mathematics to describe reality, mathematics itself is not science. Mathematicians (and mathematics) don't care about the "real world".

I would like to refute this. I like real world food, for example. Erdos cared about real world coffee, too.

However, what you said about mathematics is most certainly true. There is a sincere difference between the real world and what we can come up with in mathematics.

[...]
'I think it will be easier to explain this by example. Consider random process that produces series of red points somewhere in a unit disk with uniform probability density. The probability of the event that the next point will coincide with any point A of the disk is equal to 0.

However, after the event occurs, some point of the disk will be red. At that instant, an event with probability 0 has happened.

Actually, all events that happen in such random process are events that have probability 0.

So "event has probability 0" does not mean "impossible event".

Similarly, "probability 1" does not mean "certain event". Consider probability that the red point will land at point with both coordinates irrational. This can be shown to be equal to 1 in standard measure theory. However, there is still infinity of points that have rational coordinates, and these can happen - they are part of the disk.
[...]

I think parts of this are right because you don't know what you are saying.

A probability says nothing about certainty. It can imply "likelihood" (kind of), but it does not say that something will happen. At least, that's my view.

From your perspective, as a physicist, a small area (I can't bring myself to say "point" ) in the disk will have a positive area, and thus the probability of a random small area on that unit disk being chosen for redness is the area of that small area divided by the area of the disk.

 

[D H]

The article also points out that while it is possible to define the uniform probability distribution over [0,1] (actually it talks about (0,1), but I'll let that ride), any attempt to sample without bias (in finite time) from that set is "doomed to failure".

You misread again. Here's what the article does say:

If we try to sample uniformly at random from the set of all integers (or the set of real numbers, for that matter) we are doomed to complete failure.​

The concept of a uniform distribution over the integers, or over the reals, or over any set with infinite measure does not make a lick of sense. The concept of a uniform distribution over a finite measure subset of the reals makes an immense amount of sense. There's a huge, huge difference between a set with finite measure and one with infinite measure.

 

[Stephen Tashi]

The concept of a uniform distribution over a finite measure subset of the reals makes an immense amount of sense.

Not always. There is no uniform distribution (using Lebesgue measure) over the rational numbers in [0,1]. Yet we postulate that there is a uniiform distribution over all the reals in [0,1].

 

[lavinia]

Perhaps from a physicist's point of view, it "will" converge to zero, but I think the mathematical theorem only says that it converges to zero "with probability 1", so the dragon rears its head again.

Jano L is correct (this being the mathematics section); the statement "The probability of statement A being true is 1" does not allow one to infer that "Statement A is true" if we are using the usual axioms for probability theory. Nor does the statement "Statement A is true with probabilty 0" allow us to infer "Statement A is false". ( People have 'probably' invented logics and axioms that attempt to sort out the relation between the probability of a statement and its definite truth or falsity and make explicit the distinction that D H mentions between events that are impossible and events that have zero probability. I'm only saying that the standard approach to probability theory does not deal with such things.)

Good point. So here is a question. It seems that the Central Limit Theorem says that the distribution of the averages in this case - let's assume that the theorem applies - should converge to a normal distribution with zero variance. what does that distribution actually look like?

 

[bhobba]

From what I read, the Kochen-Specker theorem appears to make an assumption of 0 and 1 as being the only possible outcomes. Correct me if I'm wrong. If that's the case, your concern over almost never versus never doesn't apply.

I agree. It was Jano L that raised the issue - I thought it not applicable as well. But I wanted to explore the issue with those more into probability theory itself - hence my post.

Thanks
Bill

 

[bhobba]

If you flip a coin an infinite amount of times, the probability that you will flip heads every time is 0. It is certainly a possible outcome, though.

I am unsure of exactly how you would know that since flipping a coin an infinite number of times is not possible. But what you can say is the expected proportion of times you get all heads decreases towards zero as the number of flips increase.

Thanks
Bill

 

[bhobba]

While I basically agree with this and think that we should be considering more than just "real world", it comes to mind that we must at least consider the real world because I don't see how you could define "certainty" mathematically.

Maybe that the situation can be modeled with an event space of just one element. This means of course it has probability of 1 and is certain. Just a thought.

In the ensemble interpretation used in QM that would mean every element of the ensemble would be the same. Selecting any element from such an ensemble will give that outcome with dead certainty - meaning since its the only one that must be it.

However this does seem to highlight issues with applying probability theory.

I seem to recall, when I studied probability models, Ross's text - Introduction To Probability Models was used. He mentioned while abstract probability theory is an important area to add rigor to the foundations of the subject, it was still important yo have an intuitive feel for what's going on in applying it.

Thanks
Bill

 

[bhobba]

Dumb post, I didn't take the 2 seconds to reallize the difference between integers and reals.

It's not dumb in one sense - dealing with infinite sets is an issue.

In the thread this started from, over on Quantum Physics, I gave a probabilistic model of a particles position that only involved 0's and 1's. It used the Dirac probability measure (physicists simply call it the Dirac Delta function but I thought I would switch over to probability jargon here) to model a particle being at a specific point for sure. The fact you have to resort to distribution theory highlights the tricky issues it has.

Thanks
Bill

 

[Stephen Tashi]

It seems that the Central Limit Theorem says that the distribution of the averages in this case - let's assume that the theorem applies - should converge to a normal distribution with zero variance. what does that distribution actually look like?

I don't think the Central Limit theorem says anything about a normal distribution with zero variance. The question of convergence of a series of distributions is a special case of the convergence of a series of functions and several different types of convergence are defined. So we'd first have to pick which kind of convergence we want to talk about.

 

[marmoset]

Ed Jaynes discusses some of these issues in Appendix B on 'Mathematical formalities and style' in his book 'Probability Theory: The Logic of Science', available here http://omega.albany.edu:8008/JaynesBook.html.

 

[HallsofIvy]

Quote by Dmobb Jr.

I don't see how you could define "certainty" mathematically.

Quote by D H

That is almost certainly an event with a probability of 1.

So certainty is almost certain?

 

[Tamroes]

This is an interesting question as if it is a Probability then the range should go from "Near zero" to One and not from Zero to One. This is where the Quants made the error and then came up with the "Hundred Year Storm" as an excuse. Ito's Calculus is really Physics; (ballistic trajectory) and shouldn't be used in Finance. (just my opinion)