Required math: (very simple) probability & calculus
Required physics: none
This interesting little problem serves well to illustrate the notion of a probability density and its application to an experiment which can be done at home. It is known as Buffon’s needle, since it is believed that Georges-Louis Leclerc, Comte de Buffon, first posed the problem in the 18th century.
Suppose we have a needle of length and we drop this needle onto a sheet of paper on which there are a number of parallel lines spaced a distance apart. What is the probability that the dropped needle will cross a line?
One way of analyzing this problem is to begin by considering the needle as the hypotenuse of a right triangle, with sides parallel and perpendicular to the parallel lines. Then if the needle makes an angle with the lines, the sides of this triangle are for the parallel side and for the perpendicular side. The side of the triangle parallel to the lines is of no interest here; what we are interested in the perpendicular component.
To see this, suppose we placed a needle of length on the paper in such a way that it was perpendicular to the lines. Such a needle would cover a fraction of the distance between two adjacent lines. Thus the probability that a line would cross this specially dropped needle is just . (Note that if the probability is 1, since such a needle covers the entire distance between the lines.)
Therefore, the probability that a needle that makes an angle with the lines crosses a line is .
How likely is the needle to drop at an angle ? Since is a continuous variable, we need a probability density, rather than just a simple probability. Assuming that the needle is equally likely to drop at any angle between 0 and , the density must be a constant, and must integrate to 1 over the range of valid , so it must be .
The probability that a randomly dropped needle crosses a line is therefore
Besides being another of those curious situations where pops up unexpectedly, this result offers the possibility of an interesting way of whiling away a rainy afternoon. Simply by dropping a needle repeatedly onto lined paper, you can do an experiment that will determine the value of , since
OK, you would need to drop a lot of needles to get a decent value of , but it’s interesting that there is such a simple method for getting even an approximate value of without using circles, triangles, angles, measurement or anything more than just counting.