  We provide a condition on a set of directions $\Omega \subset \mathbb{S}^1$
ensuring that the associated directional maximal operator $M_\Omega$ is
unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. The techniques
of proof extend ideas of Bateman and Katz involving probabilistic construction
of Kakeya-type sets involving sticky maps and Bernoulli percolation.
