Can any physical event "almost surely" not happen? 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?
 A: 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.
