  Formalised libraries of combinatorial mathematics have rapidly expanded over
the last five years, but few use one of the most important tools: probability.
How can often intuitive probabilistic arguments on the existence of
combinatorial structures, such as hypergraphs, be translated into a formal
text? We present a modular framework using locales in Isabelle/HOL to formalise
such probabilistic proofs, including the basic existence method and first
formalisation of the Lov\'asz local lemma, a fundamental result in probability.
The formalisation focuses on general, reusable formal probabilistic lemmas for
combinatorial structures, and highlights several notable gaps in typical
intuitive probabilistic reasoning on paper. The applicability of the techniques
is demonstrated through the formalisation of several classic lemmas on the
existence of hypergraphs with certain colourings.
