My problem is the following. In the middle of an interval $L$ of length $l$ is placed an interval of length $k$ (meaning the interval stretches from $[\frac{l}{2}-\frac{k}{2}, \frac{l}{2}+\frac{k}{2}]$ ). Then we randomly create $n$ intervals, $I_1$,$I_2$, ..., $I_n$ of length $m < k$ whose centers are distributed uniformly and independently on the interval $L$.
What I would like to compute is $P$($[\frac{l}{2}-\frac{k}{2}, \frac{l}{2}+\frac{k}{2}]$ $\subseteq$ $\bigcup_{i=1}^n I_i$)
I have no idea how to express the union of the $I_i$s so that I can get to the answer if it exists (it's not coming from an exercise so I don't know if there's an analytical solution to that question actually). Any help would be appreciated. Thanks.