There are many examples in fractal geometry/dynamical systems (including of course the one in the OP).
For example, in general it is very hard to compute the dimension (any kind of fractal dimension) of a set invariant under a nonconformal dynamical system. However adding randomness makes the situation much easier. See for example the paper "Hausdorff dimension for randomly perturbed self-affine attractors" by Jordan, Pollicott and Simon.
A related example concerns Bernoulli convolutions and more general self-similar measures with overlaps. Very little is known for specific cases, but the more randomness one adds the easier the situation becomes. See for example the paper "Absolute continuity for random iterated function systems with overlaps" by Peres, Simon and Solomyak, in which they establish absolute continuity of a family of random measures, whose deterministic counterpart includes a measure whose absolute continuity was asked by Sinai motivated by connections with the Collatz conjecture. Namely, Sinai asked about the absolute continuity of the distribution $\mu_a$ of the random series $$ 1+Z_1+Z_1 Z_2+Z_1 Z_2 Z_3+\ldots, $$ where $P(Z_i=1+a)=P(Z_i=1-a)=1/2$ for some $a\in (0,1)$. (The authors of the cited paper show that if $a>\sqrt{3}/2$ then $\mu_a$ is singular, while if $a<\sqrt{3}/2$, then a variant of $\mu_a$ in which $Z_i$ has a multiplicative random error, is absolutely continuous.)
