Problem 16

The Buffon Needle Problem (1777)

Problem. A needle of length l is thrown at random on a plane on which a set of parallel lines separated by a distance d (>l) have been drawn. What is the probability that the needle will intersect one of the lines?

Solution. Consider the diagram in Fig 16.1. Let the distance between the midpoint of the needle to the nearest line be Y (0 < Y < d/2), and the acute angle between the needle and the horizontal be Φ (0 < Φ < π/2). We assume that and that . It is also reasonable to assume that Y and Φ are statistically independent. Therefore, the joint density of Y and Φ is

The needle will intersect one of the lines if and only if (see Fig. 16.1). The probability of this event is the double integral of the joint density over the area A (see Fig. 16.2):

Hence the required probability is .

# 16.1 Discussion

A simpler solution than the previous ...

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