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Does a Probability of 1 Really mean a Dead Cert

  1. May 30, 2013 #1

    bhobba

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    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 cant 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
     
    Last edited: May 30, 2013
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  3. May 30, 2013 #2

    pwsnafu

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    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.
     
  4. May 30, 2013 #3

    Stephen Tashi

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    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.)


    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.
     
    Last edited: May 30, 2013
  5. May 30, 2013 #4

    bhobba

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    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.

    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
     
  6. May 31, 2013 #5
    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.
     
  7. May 31, 2013 #6
    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.
     
    Last edited: May 31, 2013
  8. May 31, 2013 #7

    Jano L.

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    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.
     
  9. May 31, 2013 #8

    D H

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    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".



    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.
     
  10. May 31, 2013 #9

    lavinia

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    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.
     
  11. May 31, 2013 #10

    Stephen Tashi

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    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.)
     
  12. May 31, 2013 #11
    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.
     
  13. May 31, 2013 #12

    Stephen Tashi

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    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.
     
    Last edited: May 31, 2013
  14. May 31, 2013 #13
    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.
     
  15. May 31, 2013 #14
    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.
     
  16. May 31, 2013 #15
    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.
     
  17. May 31, 2013 #16
    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.
     
  18. May 31, 2013 #17

    D H

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    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".
     
  19. May 31, 2013 #18
    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.
     
  20. May 31, 2013 #19
    Last edited: May 31, 2013
  21. May 31, 2013 #20

    D H

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    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.
     
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