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Buffon's needle experiment is a very old and famous random experiment, named after the great French naturalist and historian Georges-Louis Leclerc, Compte de Buffon, who lived from 1707 to 1788. The experiment consists of dropping a needle on a hardwood floor. The main event of interest is that the needle crosses a crack between floorboards. Strangely enough, the probability of this event leads to a statistical estimate of the number π!
Our first step is to define the experiment mathematically. We idealize the physical objects by assuming that the floorboards are uniform and that each has width 1. We will also assume that the needle has length L < 1 so that the needle cannot cross more than one crack. Finally, we assume that the cracks between the floorboards and the needle are line segments.
When the needle is dropped, we want to record its orientation relative to the floorboard cracks. One way to do this is to record the angle x that the top half of the needle makes with the line through the center of the needle, parallel to the floorboards, and the distance y from the center of the needle to the bottom crack. Thus the sample space (the set of all possible outcomes) of the experiment is
S = [0, π) × [0, 1] = {(x, y): 0 ≤ x < π , 0 ≤ y ≤ 1}
Our main modeling assumption is that the needle is tossed "randomly" on the floor. This means that we have no reason to prefer one portion of S over another, and thus mathematically, the probability of an event A should be proportional to the area of A. Therefore we give S the uniform probability distribution (this is the continuous analog of the probability distribution used in the birthday and poker problems):
P(A) = area(A) / area(S) for A 
S.
Our main interest is in the event C that the needle crosses a crack between the floorboards. Once again, it's actually easier to describe the complimentary event that the needle does not cross a crack. From simple trigonometry note that
Cc = {(x, y): (L / 2) sin(x) < y < 1 - (L / 2) sin(x)}
An easy exercise in calculus now shows that area(Cc) = 2L − 1, and therefore P(C) = 2L / π
The convergence of the relative frequency of an event (as the experiment is repeated) to the probability of the event is a special case of the law of large numbers. Thus, if we run Buffon's needle experiment a large number of times the proportion of crack crossings should be about the same as the probability of a crack crossing. More precisely, we will denote the number of crack crossings in the first n runs by Nn. (Note that Nn is actually a random variable for the compound experiment that consists of repeated, independent replications of the basic needle experiment). In any case, if n is large, we should have
Nn / n ~ 2L / π and hence π ~ 2Ln / Nn.
This is Buffon's famous estimate of π.
The applet for Buffon's needle experiment shows the physical outcome of the experiment in the first graph panel. The second graph panel shows the sample space of the experiment, with the crack-crossing event B outlined in blue. (More precisely, Cc is the region between the curves, while C consists of the region above the upper curve and the region below the lower curve.) For each run of the experiment, the outcome (x, y) is recorded as a red dot in the sample space (so this graph is actually a scatterplot). Additionally, x and y are recorded in the first table on each update. An indicator variable I indicates whether or not C occurs. This variable is also recorded in the first table on each update. The probability density function of I, which simply gives the probability of C and Cc, is shown in blue in the third graph panel. When the simulation runs, the empirical density function of I, which simply gives the relative frequency of C and Cc, is shown in red. The second table displays the same information in numerical form. Finally, the last graph panel shows π in blue and the estimate of π in red. The last table gives the same information in numerical form.
1.
Run Buffon's needle experiment with the default settings and watch the outcomes being plotted in the sample space. Note how the points in the scatterplot seem to fill the sample space S in a uniform way.
2.
In the simulation of Buffon's needle experiment, vary the needle length L with the scroll bar and watch how the events C and Cc change. Run the experiment with various values of L and compare the physical experiment with the points in the scatterplot. Note the apparent convergence of relative frequency of C to the probability of C.
3.
Run Buffon's needle experiment repeatedly with needle lengths L = 0.3, 0.5, 0.7, and 1. In each case, watch the estimate of π as the simulation runs.
The estimate of π tends to improve as the needle length increases. This is not easy to see mathematically, but the applet illustrates this fact empirically.
4. In the Buffon's needle experiment, set the update frequency to 100. Run the simulation 5000 times each with L = 0.3, L = 0.5, L = 0.7, and L = 1. Note how well the estimator seems to work in each case.
Finally, we should note that as a practical matter, Buffon's needle experiment is not a very efficient method of approximating π. According to Richard Durrett in The Essentials of Probability (Duxbury Press, 1994), to estimate π to four decimal places with L = 1 / 2 would require about 100 million tosses!
5.
Run Buffon's needle experiment with an update frequency of 100 until the estimates of π seem to be consistently correct to two decimal places. Note the number of runs required. Try this for needle lengths L = 0.3, L = 0.5, L = 0.7, and L = 1 and compare the results.
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