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I asked a Question (Why do we say "almost surely" in Probability Theory??) about what exactly "almost surely" means and got some really good, helpful answers. The examples of events that almost surely cannot happen were clear, but I noticed that they were also all physically unrealizable events.

A coin almost surely will not lands heads an infinite number of times. A random selection from a uniform distribution almost surely will not equal exactly 1/2. Every person who enters a raffle that infinitely many people buy tickets for almost surely will lose. All true. But of course, it's not possible to flip a coin an infinite number of times, it's not possible to express (let alone choose) a truly random number since most numbers are irrational, and there'd be no way to notify the Infite Raffle winner because you couldn't publish his infinite winning number.

So, is there any physically realizable process to which any specific and specifiable result almost surely cannot happen?

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Jerry Guern
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    Getting a 7 on a six-sided dice? – Gummy bears Sep 20 '15 at 05:13
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    @Gummybears: It almost surely does not occur. But is not it even impossible to occur? – Yes Sep 20 '15 at 05:20
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    Check this out: http://www.jstor.org/stable/2946572?seq=1#page_scan_tab_contents. Of course, you could argue that it's not really physical because the author does not account for relativity. But still cool. Stuff like this pops up in physics, like the probability of swinging a pendulum and having it get stuck exactly at the top. – Plutoro Sep 20 '15 at 05:21
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    @AlexS That's a good one. Of course, Quantum Mechanics tells us there's no such thing as the pendulum getting stuck exactly at the top, and anyway, to get stuck it only needs to stop close enough to the top to not overcome van der Waals friction, blah blah blah. – Jerry Guern Sep 20 '15 at 05:29
  • @GudsonChou Well in this world, if we account for quantum physics, nothing is really 'impossible', is it? – Gummy bears Sep 20 '15 at 05:29
  • @Gummybears: Ah, quantum world. Yes, I overlooked that point of view. :) – Yes Sep 20 '15 at 05:30
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    @Gummybears Even in a QM world, getting a 7 on a six-die is impossible. If you did roll a 7 or the dice turned into an elephant, you'd consider the roll invalid. – Jerry Guern Sep 20 '15 at 05:32
  • @JerryGuern QM and GR are wrenches in the works. But to exrend the example, classically, any equilibrium in an ODE has a stable manifold of measure $0$ will almost surely never be reached. – Plutoro Sep 20 '15 at 05:33
  • @JerryGuern We may have extra dots appear on a face. See, it depends on how you look at it. Why should I consider the roll invalid? – Gummy bears Sep 20 '15 at 05:33
  • Even in the quantum world some things are impossible. It is impossible to measure the spin of an electron to be anything other than $\pm\hbar/2$. But some things are almost certainly not going to happen, like measuring the position of a particle (bound in any physical potential) to be at exactly $1$. – Plutoro Sep 20 '15 at 06:09
  • You say that a coin will almost surely not land heads an infinite number of times (if you throw it infinitely often?). This is false. But perhaps more related to your question: If you prescribe any outcome $(x_n)_n \in {H,T}^\Bbb{N}$, then every such outcome has probability zero. Nevertheless, one of them will occur. – PhoemueX Sep 20 '15 at 11:41
  • @PhoemueX Just asserting that something is false without explaining yourself is not very useful. – Jerry Guern Sep 20 '15 at 12:23
  • @JerryGuern The OP meant that a coin will almost surely land heads an infinite number of times, assuming it is thrown an infinite number of times. The odds of it being thrown a finite number of times is clearly 0, but still within the probability space. – Nicholas Pipitone Sep 20 '15 at 12:51
  • @JerryGuern: Yeah, sorry. I now realize that maybe the wording of the statement is just ambiguous. If you throw the coin an infinite number of times, you will almost surely get heads infinitely often and tails infinitely often, this is a consequence of the Borel Cantelli lemma. – PhoemueX Sep 20 '15 at 13:02
  • @PhoemueX Ah, okay. Yes, I meant that if we toss the coin infinitely many times we almost surely will not get all heads. – Jerry Guern Sep 20 '15 at 13:24
  • @JerryGuern : I followed your coin toss and uniform random variable example (those have well defined probability meanings), but not your raffle example. It is not possible to define a uniform mass function over a countably infinite set. So, the raffle win probabilities for each person must be nonuniform (such as $(1/2)^n$ for person $n$) and so it is indeed possible for a particular person to win with positive probability. – Michael Sep 20 '15 at 13:35
  • @Michael You answered your own objection: "It is not possible to define a uniform mass function over a countably infinite set" Right! Because the mass function would just go to zero. Every player would almost surely not win because his probability of winning is zero. – Jerry Guern Sep 20 '15 at 13:41
  • @JerryGuern : My point is that the situation of the raffle cannot even be constructed within the (possibly imaginary) world of probability, while the coin toss and uniform examples can. Unless, for example, you assume there are an uncountably infinite number of participants and go back to the uniform random variable over $[0,1]$ example. – Michael Sep 20 '15 at 13:42

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From Wikipedia:

Throwing a dart[edit] For example, imagine throwing a dart at a unit square (i.e. a square with area 1) wherein the dart will impact exactly one point, and imagine that this square is the only thing in the universe besides the dart and the thrower. There is physically nowhere else for the dart to land. Then, the event that "the dart hits the square" is a sure event. No other alternative is imaginable.

Now, notice that since the square has area 1, the probability that the dart will hit any particular sub-region of the square equals the area of that sub-region. For example, the probability that the dart will hit the right half of the square is 0.5, since the right half has area 0.5.

Next, consider the event that "the dart hits the diagonal of the unit square exactly". Since the area of the diagonal of the square is zero, the probability that the dart lands exactly on the diagonal is zero. So, the dart will almost never land on the diagonal (i.e. it will almost surely not land on the diagonal). Nonetheless the set of points on the diagonal is not empty and a point on the diagonal is no less possible than any other point, therefore theoretically it is possible that the dart actually hits the diagonal.

The same may be said of any point on the square. Any such point P will contain zero area and so will have zero probability of being hit by the dart. However, the dart clearly must hit the square somewhere. Therefore, in this case, it is not only possible or imaginable that an event with zero probability will occur; one must occur. Thus, we would not want to say we were certain that a given event would not occur, but rather almost certain.

I don't know if you consider this a physical event, but you can try in a computer program to generate two numbers in between 0 and 1 "randomly". Will they ever be equal? No, almost surely.

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    In this case using computers might not be the best option because their representation of numbers is in $\mathbb{Q}$, so you wouldn't put the same probability measure onto that set. –  Dec 04 '15 at 10:34