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  "body": "\n## Summary\n\nThe probabilistic method proves that an object with a given property exists.\nIt chooses an object at random and shows that it has the property with positive probability.\nIt is a nonconstructive method: it usually gives no explicit example.\nErdős made it a main tool of combinatorics, starting with lower bounds for [[Ramsey numbers|0K9QVPJYJF]].\nIts tools include [[linearity of expectation|0P7PTC8DDE]] and [[Boole's inequality|0EGSEE9J9R]].\nMore refined tools are the [[Lovász local lemma|35YQ94FNV1]] and concentration inequalities.\n\n## Explanations\n\n### L1 human\n\nTo show that something exists, it is enough to show that a random choice finds it with a chance above zero.\nFor example, colour the edges of a complete graph red or blue by fair coin flips.\nIf the expected number of single-coloured large cliques is below $1$, some colouring has none.\nSo a good colouring exists, even though the argument does not tell you which one it is.\n",
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