In the study of the observability of the wave equation (here on $(0,T)\times \T^d$, where $\T^d$ is the d-dimensional torus), a condition naturally emerges as a sufficient observability condition. This condition, which writes $\ell^T\left(\omega\right)>0$, signifies that the smallest time spent by a geodesic in the subset $\omega\subset \T^d$ during time $T$ is non-zero. In other words, the subset $\omega$ detects any geodesic propagating on the d-dimensional torus during time $T$. Here, the subset $\omega$ is randomly defined by drawing a grid of $n^d$, $n\in\N$, small cubes of equal size and by adding them to $\omega$ with probability $\varepsilon>0$. In this article, we establish a probabilistic property of the functional $\ell^T$: the random law $\ell^T\left(\omega_\varepsilon^n\right)$ converges in probability to $\varepsilon$ as $n \to + \infty$. Considering random subsets $\omega_\varepsilon^n$ allows us to construct subsets $\omega$ such that $\ell^T\left(\omega\right)=|\omega|$.