Probability theory says that if an event $E$ is certain to happen, then $P(E)=1$ which makes sense. Similarly, an impossible event has probability $0$.
What surprised me is the fact that you can still find mathematical texts (notice that this paper comes from a renowned American university) that say the converse are also true, namely:
$P(E)=1 \implies E\quad$ is certain to happen
$P(E)=0 \implies E\quad$ can't happen.
Now let's consider the second case. Let's say I'm choosing a point randomly from the interval $[0,2]$. Even though it's possible for every particular point to be chosen (I can easily choose $2$ or $0.5$), the calculated probability for randomly choosing that particular point is $0$. But I have chosen a point, right? Thus it is can happen.
In this case, the probability should be considered as a limiting value. When $P($the randomly chosen number equals $1$$)=0$, it should be understood as the limit of the number of times I've chosen $1$ divided by the number of trials. As the number of trials increases, this fraction approaches $0$ - but it doesn't have to be $0$ at any point during that process.
Similarly, in the first case, I might consider of an event that the randomly selected point from $[0,1]$ is from interval $[0,1)$. The measures of those sets are identical, so the probability equals $1$. Does it mean I will certainly select a point from $[0,1)$? Of course not, because $1$ can be chosen. Thus the event is not certain to happen.
Is there anything wrong with reasoning above? Why are so many people convinced the two implications are true?