TLDR
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Buffon’s Needle result (expected line crossings = 2L/πW) is derived elegantly without integrals by bending a needle into a noodle, then a circle.
Key Takeaways
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Linearity of expectation alone proves that expected crossings scale linearly with curve length, no independence assumption needed.
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Any rectifiable curve dropped on ruled lines has expected crossings proportional only to its total length, not its shape.
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The constant c = 2/πW is pinned by one geometric fact: a circle of radius W/2 crosses exactly one ruled line twice, giving E = 2.
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The noodle argument sidesteps double integrals and makes the hidden circle in the π explicit through polygonal approximation and limits.
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Rigorous treatment requires Haar measure for “random” orientation and continuity of f(L); the post cites Klain and Rota (1997) for full details.
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