BUFFON’S NEEDLE PROBLEM Abby Yinger Probability theory and stochastic processes for additional applications Geometric probability and stochastic geometry

BRIEF HISTORY  Buffon’s Needle problem is one of the oldest problems in the field of geometrical probability.  It was first proposed in 1733 by Georges-Louis Leclerc, Comte de Buffon and later solved by Buffon in 1777.

WHAT IS THIS PROBLEM?  Find the probability that a needle of length t will land on a line, given a floor with equally spaced parallel lines a distance d apart.

THE SOLUTION  To solve this problem we need to look at three different cases:  Case One: Where the object being thrown (a needle in Buffon's case) is equal to the distance between the lines  Case Two: Where the object being thrown is smaller than the distance between the lines.  Case Three: Where the object being thrown is larger than the distance between the lines.

CASE ONE  Let L be the length of the needle and therefore the distance between the lines in the floor. Suppose we throw the needle at an angle considered to be between 0 and by symmetry (if the angle is greater than, such as an angle +, the needle will be in the same position as when thrown with an angle itself). We can draw a line parallel to the lines on the floor, passing through the center of the needle. Let d be the distance from the center of the needle to the closest line. As shown in Figure 1, if ≤ the needle hits the line, but if this is not true the needle does not hit the line.

CASE ONE CONT.

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