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Let's pick a random real number from $[0,1]$. What's the probability we picked number $1$?

It can't be a nonzero $\alpha$. Otherwise we could pick $n$ numbers from $[0,1]$ so that $0<\frac1{n}<\alpha$. One of those $n$ numbers must have a probability of being picked less than $\alpha$, but we picked randomly, so all probabilities should be equal.

But if the probability is $0$, does it mean that it's impossible to pick $1$?

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    $\begingroup$ The probability of picking any particular number is zero (assuming a continuous distribution). That means it is improbable, not impossible. Clearly when you pick a number it is possible. $\endgroup$ – copper.hat Oct 8 '14 at 18:00
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    $\begingroup$ There is a small difference between an event having probability zero and an event being impossible. Which is why a mathematician would say that you almost certainly wouldn't happen to chose the number $1$. $\endgroup$ – Arthur Oct 8 '14 at 18:14
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No, zero probability does not mean that event is impossible. Every time you pick a random number from $[0,1]$, you have an infinite set of possible outcomes, so probability of picking a number from any finite subset of $[0,1]$ is equal to zero. However, impossible event is defined otherwise. It's an event which can't happen due to experiment conditions. In this case it's picking any number outside of $[0,1]$.
Thus, impossible and zero-probable events have the same probability of $0$, however impossible events never happen, while zero-probable events are possible. It follows from the fact that a length of segment $[1,1]$ (single point) is $0$, however point $1$ (as well as any other point) exists.

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    $\begingroup$ not only any finite subset, but also any countably infinite subset, and any (possibly uncountable) set of measure 0. $\endgroup$ – genisage Oct 8 '14 at 18:23
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As far as I know, "impossible event" is not standard mathematical terminology. Inasmuch as definitions are arbitrary, you can define it to mean "empty event" or "probability zero event" as you choose; either way, it does not fill a need, because there is already a name for it.

In pure mathematics, it's just a matter of definition. But I suppose your question is about applied mathematics: if the mathematical theory of probabilities is applied to real world events, can a probability zero event ever happen? Is that what's troubling you? The answer depends on what physical mechanism you are using to choose a random number in $[0,1]$ and how you decide whether the chosen number is exactly equal to $1$. I believe that, however you set up the experiment, hitting $1$ right on the nose will either be impossible, or else will have some nonzero probability. But that's just my opinion.

"Any clod can have the facts, but having opinions is an art."

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Probability zero means not likely to be true or to happen. It never says it is never likely to be possible or to be true. You can prove this by simple geometric probability, Just it is equivalent of picking a point from a line segment which is zero.

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