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Buffon's constant is . Its decimal expansion is 0.63661977236758... (see A060294) and its continued fraction is
Like many other numbers involving π, Buffon's constant can be expressed as an infinite sum or product:
It is also the limit , 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 that a needle of length will randomly land on a line, given a floor with equally spaced parallel lines at a distance apart, assuming that the angle and the position of the fallen needle are independently and uniformly random, is .
Proof. If the needle always fell perpendicular (angle radians) to the parallel lines, we would have . So we have
as specified by the theorem. □