Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with probability 1. Put $I_i=[x_i, x_{i-1}]$.

Next we pick $n$ random numbers $y_1, \ldots, y_n$ in $[0, 1]$ independently with respect to the uniform distribution. Let $J$ be the union of all $I_i$, for which there exists some $y_j$ with $y_j\in I_i$. Again it is obvious that $|J|\rightarrow 1$ as $n\rightarrow\infty$ with probability 1. My question is how quickly does $1-|J|$ decay? I would expect that the distribution of $1-|J|$ is not very concentrated, so I am interested both in an estimate for the expected value as for median and extreme values.

The above problem occurs quite naturally in the analysis of algorithms, so I would expect that this problem has been addressed by someone. Several random structures have parts of sizes following the distribution of the $x_i$, and picking $y_i$ corresponds to picking random points in a random structure and studying the component containing this point. The question is then how much of the whole structure has been left unexplored.