What are the examples of situations where "randomizing" a problem (or some part of it) and analyzing it using probabilistic techniques yields some insight into its deterministic version?

An example of what I have in mind: it is a well-known conjecture that the Hausdorff dimension of the graph of Weierstrass function (everywhere continuous, nowhere differentiable) is given by a certain simple formula, depending on the amplitudes and phases of the cosines in the series. This is still open; however, in the paper "The Hausdorff Dimension of Graphs of Weierstrass Functions" Hunt proved that if you add a uniformly distributed independent random "noise" to each phase, the conjectured formula holds with probability 1. So while the "randomized" approach does not solve the original problem, it somehow lends credibility to the original conjecture and thus gives us some insight about the problem.