# Zero probability? [duplicate]

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 $1$?

• 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. – copper.hat Oct 8 '14 at 18:00
• 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 $1$. – Arthur Oct 8 '14 at 18:14

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.
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.