  Given a sequence of random variables $\left\{ X_k : k \geq 1\right\}$
uniformly distributed in $(0,1)$ and independent, we consider the following
random sets of directions $$\Omega_{\text{rand},\text{lin}} := \left\{
\frac{\pi X_k}{k}: k \geq 1\right\}$$ and $$\Omega_{\text{rand},\text{lac}} :=
\left\{ \frac{ \pi X_k}{2^k} : k\geq 1 \right\}.$$ We prove that almost surely
the directional maximal operators associated to those sets of directions are
not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$.
