Definition 1. A sequence $(x_n)$ of real numbers on $[0,1]$ is said to be equidistributed on $[0,1]$ if for any interval $[a,b] \subseteq [0,1]$, the sequence $\frac{\{x_1,\ldots,x_n\} \cap [a,b]}{n}$ converges to $b-a$ as $n \to \infty$.
Definition 2. A sequence $(x_n)$ of real numbers on $[0,1]$ is said to be uniformly equidistributed (or well-distributed) on $[0,1]$ if for any interval $[a,b] \subseteq [0,1]$, the sequence $\frac{\{x_{k+1},\ldots,x_{k+n}\} \cap [a,b]}{n}$ converges to $b-a$ uniformly in $k$ as $n \to \infty$.
For some discussion of these notions, see for example the wikipedia page: https://en.wikipedia.org/wiki/Equidistributed_sequence
Question 1. What is the probability that a random sequence on $[0,1]$ is equidistributed? Particularly, is it true that almost all sequences on $[0,1]$ are equidistributed?
Question 2. What is the probability that a random sequence on $[0,1]$ is uniformly equidistributed? Particularly, is it true that almost none sequences on $[0,1]$ are uniformly equidistributed?
Are these some well-known results?