A common way to generate fractal sets is by taking certain types of number expansions and considering their restricted digit sets, the sets of numbers that have such a number expansion with imposed restrictions on the corresponding digit sequence. In this talk we will consider the restricted digit sets of random Lüroth expansions, series expansions of numbers in the real interval induced by a random dynamical system of Lüroth type transformations. In particular, we will use tools from the theories of dynamical systems and iterated function systems to find computable expressions for the Hausdorff dimensions of these sets.