Given any measurable subset $\omega$ of a closed Riemannian manifold and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. Our first main result, which is of geometric nature, states that, under regularity assumptions, $1/2$ is the maximal possible discrepancy of $\ell^T$ when taking the closure. Our second main result is of probabilistic nature: considering a regular checkerboard on the flat two-dimensional torus made of $n^2$ square white cells, constructing random subsets $\omega_\varepsilon^n$ by darkening cells randomly with a probability $\varepsilon$, we prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$. We discuss the consequences in terms of observability of the wave equation.