Imagine the plane covered with a grid of parallel lines 1 unit apart. Toss a needle of length L onto the plane so that it lands in a random position with a random orientation. What is the probability that the needle lands crossing one of the lines? Clearly the solution is a function of L, the probability being near 1 for a long needle and near zero for a short needle. The answer is 2/Pi for a needle of length 1.
Think of the needle as a vector. The configuration of the vector can be described by two variables:
Each of these variables is assumed to be uniformly distributed, that is, any
configuration is just as likely as any other to occur. There is no loss of generality
in assuming the line to the left of the vector is x = 0. Clearly the y coordinates
are not relevant to the problem. These assumptions give rise to Configuration Space with
each point of Configuration Space determining one possible way the needle can land.
If the coordinates of the tail of the vector are (x,y), then the coordinates of its head are
(x + cos(theta), y + sin(theta)). Thus the needle crosses (or touches) a line whenever
x + cos(theta) >=1 orThus the part of Configuration Space corresponding to the needle crossing a line is the blue part consisting of 3 pieces A1, A2 and A3. The probability that the needle crosses a line is the ratio of the sum of the areas of A1, A2 and A3 to the area of Configuration Space, which is 2*Pi. The areas of A1, A2 and A3 are calculated by integrals.
x + cos(theta) <= 0
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