Monte Carlo Simulations - An Introduction by

Means of Buffon’s Needle Problem

Brinda Venkataramani

February 6, 2016

10 cm

5 cm

θ

1 Introduction

The above image depicts the intersection of a matchstick, 5 cm in length, witha line drawn on a sheet of paper. The paper has several parallel lines drawnon it such that the distance between each line is 10 cm. The matchstick de-

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picted is one of several strewn about the paper. Some cross lines; others donot. While a strange experiment, Buffon’s needle experiment as it is called(for he used needles instead of matchsticks), marks an important advent in thestatistical/probabilistic analysis of apparently chaotic data.

Before getting into the mathematics of this enigmatic experiment, I will firstconvince you of its importance. This experiment was reproduced (by myself)with 163 matchsticks. Of these matchsticks, 52 crossed/contacted a line markedon this sheet of paper when randomly strewn about. Dividing these values weobtain,

162

52= 3.1346... ≈ π

Repeating this experiment several times, often with several hundreds of thou-sands of matchsticks, we find that the ratio between the total number of matches,and the number of matches crossing/contacting a line, denoted N and n respec-tively, is always an approximation of π. As expected, the greater the number ofmatchsticks used, the better the approximation. Obviously, the mathematicalsignificance of this outcome was not missed by mathematicians at the time. Thefollowing section hence serves to elucidate the logical deductions which came ofthis experiement, after which a brief section on applications follows.

References

[1] Mendivil, F., Shonkwiler, R.W. (2009). Explorations in Monte Carlo meth-ods. New York, NY: Springer LLC.

[2] Pi and Buffon’s Needle - Numberphile [Video file]. Retrieved fromhttps://www.youtube.com/watch?v=sJVivjuMfWA

2 Probability Theory

Probability while common is an abstract concept. It plays an important role inthe sciences. For example, when attempting to determine correlation betweentwo variables, it is necessary to determine the extent to which the variablesco-occur. Consider the example in which event A is observed to occur spatiallyand temporally prior to event B. While sufficient to casually state that therelationship is causal, mathematics and science rely on more mathematicallyprecise definitions of causality. That is, if and only if, A precedes B in theabsence of all other environmental factors, can A be said to cause B.

Naturally, even the most well designed experiment is flawed in some way oranother. Most kinematics experiments conducted in the classroom are flawed inthat no surface can be truly frictionless; a reaction carried out in a laboratorymay only be carried out under random pressure/temperature fluctuations; andso on. Hence, it is necessary for scientists to determine a set of rules whichdictate whether or not an experimental result is statistically significant.

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As a trivial example: Which of two statements – a) in a carefully controlledexperiment, nine of ten flowers bloomed after being pollinated artificially bloom,b) in a carefully controlled experiement, five of ten flowers bloom after beingwatered – suggest a causal relationship? Instinct tells us that statement A ismuch more likely to be true. Why? Simply because it occurs more often thennot.

This is the nature of correlation: That if an event, A, occurs, then an event,B, ensues, a majority of the time in a controlled environment. What thenis the relevance of probability to the matchstick problem? Well, what is theprobability that a matchstick will cross/contact a line?

References

[1] Mendivil, F., Shonkwiler, R.W. (2009). Explorations in Monte Carlo meth-ods. New York, NY: Springer LLC.

3 Mathematical Interpretation of the Phenomenon

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