# The Toothpick Problem

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2023-03-16 21:30:07

Take a square piece of paper. Take a toothpick of a given length. At intervals equal to the length of the toothpick, draw lines on the piece of paper. Now, randomly toss a bunch of such toothpicks so that they fall over the paper. What fraction of the toothpicks will fall in a way so that they intersect one of the lines drawn? In other words, what is the probability of a single toothpick falling over a line?

Let us first simplify the problem. Consider the simplest case when there is only one line drawn. If there is more than one line drawn, the probablity will still be the same. This is because, a toochpick has equal probability of falling over any given line. Thus, we may assume that there is only one line drawn, without loss of generality.

The problem therefore simplifies to as follows: Refer to the above figure. Suppose we have a piece of paper, over one of whose edge, a line is drawn. We must find the probability that a toothpick, of length equal to the width of the paper, falls on the line drawn.