Buffron Needle Problem

The assumption would be that the lines are parallel to the x-axis. Two variables exist in this scenario. the angle with which the needle would be tilted to the lines, Ɵ, and the distance between the closest line and the center, which would never exceed half the distance between the equidistant lines (See fig. 2 below)

The assumption is that the needle would land at a random point between the lines and the angle of inclination would be another random variable. It should further be assumed that the random variables are a result of uniform distribution. This picture depicts the needle missing the line and the needle could only be on the line if the distance closest to a line, D would be less than or equal to ½SinƟ, that is D ≤ ½SinƟ. This could be illustrated better using figure 3 below.

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From this, the values representing a hit would occur below or on the curve, D ≤ 1/2SinƟ. The probability of hitting the line would be computed as a ratio of the shaded curve to the area of the entire rectangle. The area under the curve would be computed using integrals between zero and π. This will give a result of 1. The rectangle would have an area of ½π. The probability of hitting the line would therefore be 1/ (½π) = 2/π, which is approximately 0.6366. To work out π, the number of needle drops, say N would be multiplied by two and then divided by the number of hits, N’. That is:

This would be a situation where the needle is long and its length, l, would be greater than the distance between the equidistant lines. Hence, x = l/d &gt. l. The probability that the needle would intersect a line would be computed using the expression below:

Consider the size parameter x = l/d. Taking N as the number of crossings and n as the tosses made by a short needle of size parameter x, then N would have a binomial distribution as a function of n and 2x/π.

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