Buffon's Needle Problem

Find the probability $P(\ell,d)$ that a needle of length $\ell$ will land on a line, given a floor with equally spaced Parallel Lines a distance $d$ apart.

$$P(\ell,d) = \int_0^{2\pi} {\ell\vert\cos\theta\vert\over d} {d\theta \over 2\pi} = {\ell\over 2\pi d} 4\int_0^{\pi/2} \cos\theta\,d\theta$$

$$= {2\ell\over \pi d} [\sin\theta]^{\pi/2}_0 = {2\ell\over \pi d}.$$

Several attempts have been made to experimentally determine $\pi$ by needle-tossing. For a discussion of the relevant statistics and a critical analysis of one of the more accurate (and least believable) needle-tossings, see Badger (1994).

See also Buffon-Laplace Needle Problem

References

Badger, L. ``Lazzarini's Lucky Approximation of $\pi$.'' Math. Mag. 67, 83-91, 1994.

Dörrie, H. ``Buffon's Needle Problem.'' §18 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 73-77, 1965.

Kraitchik, M. ``The Needle Problem.'' §6.14 in Mathematical Recreations. New York: W. W. Norton, p. 132, 1942.

Wegert, E. and Trefethen, L. N. ``From the Buffon Needle Problem to the Kreiss Matrix Theorem.'' Amer. Math. Monthly 101, 132-139, 1994.