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  "body": "\n## Definition\n\nLet $\\Omega$ be a nonempty finite set of objects and let $P$ be a property that objects in $\\Omega$ may have.\nA proof by the *probabilistic method* that some element of $\\Omega$ has $P$ does the following:\n\n1. It puts a probability measure $\\Pr$ on $\\Omega$, making it a [[probability space|0NFPJVJJHX]].\n2. It shows that the [[event|031TPJVS74]] $\\set{\\omega \\in \\Omega : \\omega \\text{ has } P}$ has positive probability.\n3. It concludes that this event is nonempty, so some element of $\\Omega$ has $P$.\n\nA common form of step 2 uses a random variable $X$ on $\\Omega$: if the [[expected value|0WQNRCNY3M]] satisfies $\\mathbb{E}[X] \\ge t$, then $X(\\omega) \\ge t$ for some $\\omega$.\n\n## Remarks\n\nErdős' bound for diagonal Ramsey numbers is an example.\nLet $k \\ge 3$ and $n = \\floor{2^{k/2}}$, and colour each edge of the complete graph $K_n$ red or blue independently with probability $1/2$.\nThe expected number of monochromatic copies of $K_k$ is $\\binom{n}{k} 2^{1 - \\binom{k}{2}}$, and this is less than $1$.\nSo some colouring has no monochromatic $K_k$, and the [[Ramsey number|0K9QVPJYJF]] satisfies $R(k, k) \u003e \\floor{2^{k/2}}$.\nAn argument that only shows that the event has probability at least $0$ is not a proof by this method.\nStep 2 fails there, and an event of probability $0$ may be empty.\nThe method also applies to infinite probability spaces, with the same conclusion that an event of positive probability is nonempty.\n",
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