The Needle Problem: An Experimental Determination of 'Pi'

The reference where I found this Needle problem is Dirk Struik's A Concise History of Mathematics (Dover, 1987, 4th revised edition, p.128). Introduced by Georges-Louis Leclerc in 1777, it is known as the first example of a geometrical probability. It essentially consists of throwing a needle on a plane covered with parallel and equidistant lines and counting the number of times the needle hits a line.

First a model has to be made. It is easily seen that, if the needle length is larger than the distance between
the lines, the mathematical description will be easier. (why?) One can follow the following guidelines to develop a formula for the probability of the needle hitting a line as a function of the needle length and the distance between the lines. Probability should only depend on the ratio of the two numbers. (why?)

• Assume that a plane is filled with parallel horizontal lines of uniform spacing between them. Denote the needle length by l and the distance between the lines by d. Say one corner of the needle lands a distance h away from the line that is below it. What is the probability of the needle hitting either lines (above and below) as a function of h? It is an expression involving arcsin's and pi.
• Now average that probability over all possible values of h. Note the expression depends only on l/d.
• The following integral can be useful in the computation of the average mentioned above

  integral(arcsin(x),x) = x*arcsin(x)+sqrt(1-x^2)
• Now you have a formula for the probability of the needle hitting a line that contains l/d and pi. Simply prepare a setup for measuring this probability by repeated trials and for different values of l/d. Estimate your errors and compare the measured value of pi to the known value, namely 3.1415926535897932384626433832795028841971693993751........ (Well, in this case it is rather difficult for the experimental accuracy to keep up with "theory"...)

Melih Sener - 1/8/98