pi thru the eye of Buffon’s needle

[AntonioLao]

Pi (p) is an irrational as well as a transcendental number. Unlike irrational radicals which can be solutions of different degrees algebraic polynomials, p could never be a root of any of these algebraic equations. Although traditionally it is defined as the ratio of the circumference over the diameter of any arbitrary circle, it can also be determined experimentally as the rational or improper fractional value of a uniformly distributed random variable. This is the variation offspring of the first Buffon’s needle problem of 1777, two hundreds thirty years ago.

To begin, randomly toss a needle of unit length as an idealized one dimensional line segment onto an equipotential plane ruled with parallel lines a distance D apart such that D is always greater than or equal to 1. Let j be the angle between the unit needle and the azimuth of the rulings, and let y be the distance from one end of the needle to any right-handed rule (left-handed works just as well). Both j and y are now uniformly distributed random variables which respectively take on values between (0 and 1) and (0 and 90°) or (0 and p/2). Furthermore, assuming that these variables are probabilistically independence their joint probability density is then the inverse product of p and D. Omitting calculus derivations, the analytical probability of needle intersecting any ruling is then given by 2/pD. For N repeated tosses and n is the number of successful intersection then p» 2N/nD.

[mkirkpatrick]

We can indeed strain on a gnat,while swallowing a camel Antonio,what then would be the
procedure to simplify this process so that it could be applied to a toe enhancing solution?


regards michael.

[theunify]

Eat a large Pizza with friends, order another one, start eating but leave one peice un eaten, lift up your finger and say "Eureka", all things in this world are either spherical, or seemingly parts of spheres. You can derive Pi from any random variable... that is as simple as I can explain it.

Thank you it has been a pleasure working with you.