Buffon’s Needle

1. Buffon’s Needle Todd Savage

2. Buffon's Needle Problem Buffon's needle problem asks to find the probability that a needle of length ‘l’ will land on a line, given a floor with equally spaced parallel lines a distance apart. The problem was first posed by the French naturalist Buffon in 1733 (Buffon 1733, pp. 43-45), and reproduced with solution by Buffon in 1777 (Buffon 1777, pp. 100-104).

3. Monte Carlo Method Monte Carlo methods are a class of computational algorithms for simulating the behavior of various physical and mathematical systems. They are distinguished from other simulation methods (such as molecular dynamics) by being stochastic, that is non deterministic in some manner - usually by using random numbers (or more often pseudo-random numbers) - as opposed to deterministic algorithms. A classic use is for the evaluation of definite integrals, particularly multidimensional integrals with complicated boundary conditions.

4. The Simplest Case Let's take the simple case first. In this case, the length of the needle is one unit and the distance between the lines is also one unit. There are two variables, the angle at which the needle falls (theta) and the distance from the center of the needle to the closest line (D). Theta can vary from 0 to 180 degrees and is measured against a line parallel to the lines on the paper. The distance from the center to the closest line can never be more that half the distance between the lines. The graph below depicts this situation. The needle in the picture misses the line. The needle will hit the line if the closest distance to a line (D) is less than or equal to 1/2 times the sine of theta. That is, D <= (1/2)sin(theta).

5. The Simplest Case (cont’d) In the graph below, we plot D along the ordinate and (1/2)sine(theta) along the abscissa. The values on or below the curve represent a hit (D <= (1/2)sin(theta)). Thus, the probability of a success it the ratio shaded area to the entire rectangle. The shaded portion is found using the definite integral of (1/2)sin(theta) evaluated from zero to pi. The result is that the shaded portion has a value of 1. The value of the entire rectangle is (1/2)(pi) or pi/2. So, the probability of a hit is 1/(pi/2) or 2/pi. That's approximately 0.6366197. To calculate pi from the needle drops, simply take the number of drops and multiply it by two, then divide by the number of hits, or 2(total drops)/(number of hits) = pi (approximately).

6. Mathematica Notebook

7. Buffon’s Needle Java Applet Buffon's Needle

8. Questions • Experimental Probability = (# of hits)/(total # of drops) • On the other hand, we have (approximately): • Theoretical Probability = 2/Pi • We can use this fact to conclude that: • 2(total # of drops)/(# of hits)= Pi (approximately) • 1. After 1,000 drops, how close would you expect to be to pi? 2. After 264 drops, the estimate of pi is 3.142857. This estimate is correct to within 2/1000 of the book value of pi. Will the next drop: • A. make the estimate more accurate?B. make the estimate less accurate?C. make it more or less accurate depending on whether it's a hit or miss? orD. impossible to say. • 3. What about the next 10 drops?

9. References http://www.mste.uiuc.edu/reese/buffon/buffon.html http://www.answers.com/topic/buffon-s-needle http://www.ms.uky.edu/~mai/java/stat/buff.html http://en.wikipedia.org/wiki/Monte_Carlo_method http://mathworld.wolfram.com/BuffonsNeedleProblem.html http://scienceworld.wolfram.com/biography/Buffon.html