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"][1] 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"][2] 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.)

  [1]: http://www.maths.bris.ac.uk/~matmj/random.pdf
  [2]: http://www.math.bme.hu/~simonk/papers/PSS3.pdf