Buffon's Needle Experiment

Java Applet Simulation of Buffon's needle experiment

Buffon's needle experiment is one of the most famous experiments in probability, and 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 pi!

Assumptions

Our first step is to define the experiment mathematically. Again 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. These will be the basic random variables of our experiment, and thus the sample space of the experiment is

Again, our main modeling assumption is that the needle is tossed “randomly” on the floor. Thus, a reasonable mathematical assumption might be that the basic random vector (X, Y) is uniformly distributed over the sample space. By definition, this means that

Simulation

The applet for this and the following pages is a simulation of Buffon's needle experiment. Buffon's floor and a graph of the sample space are shown. Each time you run the simulation, the needle falls on the floor, the outcome (X, Y) is plotted as a red dot in the scatterplot, and the values of the variables X and Y are recorded numerically in the table on the left. The needle length can be varied with the scroll bar from 0.1 to 1, in increments of 0.01.

1. Run the 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.