Buffon's constant

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Buffon's constant is $\scriptstyle {\frac {2}{\pi }}\,$. Its decimal expansion is 0.63661977236758... (see A060294) and its continued fraction is

${\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{31+{\cfrac {1}{1+\ddots }}}}}}}}}}}}\,$

(see A053300).

Like many other numbers involving $\pi$, Buffon's constant can be expressed as an infinite sum or product:

${\frac {2}{\pi }}=\prod _{n=2}^{\infty }{\frac {p_{n}+2-(p_{n}\mod 4)}{p_{n}}}={\frac {2}{3}}\times {\frac {6}{5}}\times {\frac {6}{7}}\times {\frac {10}{11}}\times \ldots$

(where $p_{n}$ is the $n$th prime number);

${\frac {2}{\pi }}=1+\sum _{n=1}^{\infty }(-1)^{n}(4n+1)\left({\frac {\prod _{i=1}^{n}2i-1}{\prod _{j=1}^{n}2j}}\right)^{3}=1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {3}{8}}\right)^{3}-13\left({\frac {5}{16}}\right)^{3}+\ldots$

It is also the limit $\lim _{n\to \infty }{\frac {z(n)}{\log n}}$, where $z(n)$ is the expected number of real zeros of a random polynomial of degree $n$ with real coefficients chosen from a standard Gaussian distribution.

Theorem. The probability $\scriptstyle P(l,\,d)\,$ that a needle of length $\scriptstyle l\,$ will randomly land on a line, given a floor with equally spaced parallel lines at a distance $\scriptstyle d\,\geq \,l\,$ apart, assuming that the angle and the position of the fallen needle are independently and uniformly random, is $\scriptstyle P(l,\,d)\,=\,{\frac {2}{\pi }}\cdot {\frac {l}{d}}\,$.

Proof. If the needle always fell perpendicular (angle $\scriptstyle \theta \,=\,{\frac {\pi }{2}}\,$ radians) to the parallel lines, we would have $\scriptstyle P_{\perp }(l,\,d)\,=\,{\frac {l}{d}}\,$. So we have

$P(l,d)\,=\,\left(\int _{0}^{\pi }\sin \theta \cdot {\frac {d\theta }{\pi }}\right)\cdot P_{\perp }(l,d)=\left({\frac {-[\cos \theta ]_{0}^{\pi }}{\pi }}\right)\cdot {\frac {l}{d}}={\frac {2}{\pi }}\cdot {\frac {l}{d}}\,$

as specified by the theorem. □