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The probabilistic method is a technique for proving combinatorial existenceresults by means of showing that a randomly chosen object has the desiredproperties with positive probability. A particularly powerful probabilistictool is the Lov\'{a}sz Local Lemma (the LLL for short), which was introduced byErd\H{o}s and Lov\'{a}sz in the mid-1970s. Here we develop a version of the LLLthat can be used to prove the existence of continuous colorings. We then giveseveral applications in Borel and topological dynamics.  * Seward and Tucker-Drob showed that every free Borel action $\Gamma\curvearrowright X$ of a countable group $\Gamma$ admits an equivariant Borelmap $\pi \colon X \to Y$ to a free subshift $Y \subset 2^\Gamma$. We give a newsimple proof of this result.  * We show that for a countable group $\Gamma$, $\mathrm{Free}(2^\Gamma)$ isweakly contained, in the sense of Elek, in every free continuous action of$\Gamma$ on a zero-dimensional Polish space. This fact is analogous to thetheorem of Ab\'{e}rt and Weiss for probability measure-preserving actions andhas a number of consequences in continuous combinatorics. In particular, wededuce that a coloring problem admits a continuous solution on$\mathrm{Free}(2^\Gamma)$ if and only if it can be solved on finite subgraphsof the Cayley graph of $\Gamma$ by an efficient deterministic distributedalgorithm (this fact was also proved independently and using different methodsby Seward). This establishes a formal correspondence between questions thathave been studied independently in continuous combinatorics and in distributedcomputing.

Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms