Probability of choose point in interval (0,1) Choose a point random uniformly in $(0,1)$. Then, this point divides the interval (0,1) into two sub-intervals.

*

*Compute the expected length of the interval containing a fixed point $s \in [0,1]$.


*Compute the expected distance of the randomly chosen point from $s$.
My approach: My intuition is that the expected length of the interval containing $s$ is 1/4, since on average the sub-intervals should be about length 1/2 each, and $s$ is in exactly one of them. I don't know how to mathematically prove this intuition if it's correct. I have no idea how to approach 2.

Hint. By symmetry, you can assume $s \le 1/2$. Then for random  $x$ the interval containing $s$ has length $x$ if $x > s$ and length $1-x$ if $x<s$.
