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Consider a walk in the plane made of $n$ unit steps, with directions chosenindependently and uniformly at random at each step. Rayleigh's theorem assertsthat the probability for such a walk to end at a distance less than 1 from itsstarting point is $1/(n+1)$. We give an elementary proof of this result. Wealso prove the following generalization valid for any probability distribution$\mu$ on the positive real numbers: if two walkers start at the same point andmake respectively $m$ and $n$ independent steps with uniformly randomdirections and with lengths chosen according to $\mu$, then the probabilitythat the first walker ends farther than the second is $m/(m+n)$.

(Very) short proof of Rayleigh's Theorem (and extensions)