Buffon's Needle
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The Buffon's needle problem was the first recorded use of a Monte Carlo method. These methods employ repeated random sampling to approximate a probability, instead of computing the probability directly. Monte Carlo calculations are especially useful when the nature of the problem makes a direct calculation impossible or unfeasible, and they have become more common as the introduction of computers makes randomization less laborious. | The Buffon's needle problem was the first recorded use of a Monte Carlo method. These methods employ repeated random sampling to approximate a probability, instead of computing the probability directly. Monte Carlo calculations are especially useful when the nature of the problem makes a direct calculation impossible or unfeasible, and they have become more common as the introduction of computers makes randomization less laborious. | ||
- | + | Pi is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space. Pi is an irrational number, which means that its value cannot be expressed exactly as a fraction a/b, where a and b are integers. Since π is irrational, mathematicians have been challenged with trying to determine increasingly accurate approximations. In the past 50 years especially, improvements in computer capability allow mathematicians to determine more decimal places. Nonetheless, better methods of approximation are still desired. | |
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- | A recent study conducted the Buffon's Needle | + | A recent study conducted the Buffon's Needle experiment to approximate pi using computer software. The researchers administered 30 trials for each number of drops, and averaged their estimates for pi. They estimated the improvement in accuracy as more trials were conducted. |
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Revision as of 11:21, 28 May 2010
- The Buffon's Needle problem is a mathematical method of approximating the value of pi involving repeatedly dropping needles on a sheet of lined paper and observing how often the needle crosses a line.
Buffon's Needle |
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Basic Description
The method was first used to approximate pi by Georges-Louis Leclerc, the Comte de Buffon, in 1777. Buffon apparently first tried throwing bread sticks over his shoulder and counting how often they crossed a line on his tile floor. He was intrigued by the relationship he observed between the probability of a cross and pi.
Subsequent mathematicians have used the method with needles instead of bread sticks, or with computer simulations. In the case where the distance between the lines is equal the length of the needle, an approximation of pi can be calculated using the equation
A More Mathematical Explanation
Will the Needle Cross a Line?
To prove that the Buffon's Needle experiment will give an approximation of pi, we can consider which positions of the needle will cause an intersection. The variable θ is the acute angle between 0 and made by the needle and an imaginary line parallel to the ones on the paper. Finally, D is the distance between the center of the needle and the nearest line.
Since the needle drops are random, there is no reason why the needle should be more likely to intersect one line than another. As a result, we can simplify our proof by focusing on a particular strip of the paper bounded by two lines.
We can extend line segments from the center and tip of the needle to meet at a right angle. A needle will cut a line if the green arrow, D, is shorter than the leg opposite θ. More precisely, it will cross when
See case 1, where the needle falls at a relatively small angle with respect to the lines. Visually we can see that a needle lying at such a small angle will only intersect if the center falls close to one of the parallel lines. In case 2, the needle crosses even though the center of the needle is far from both lines because the angle is so large.
The Probability of an Intersection
To approximate pi using Buffon's method, we need to know the probability of a cross. If we graph the outcomes of θ along the X axis and D along the Y, we have the sample space for the trials. In the diagram below, the sample space is contained by the dashed lines.
The sample space is useful in this type of simulation because it gives a visual representation of all the possible ways the needle can fall. Each point on the graph represents some combination of an angle and distance that a needle might occupy. We divide the area that contains combinations that will cause an intersection by the total possible positions to calculate the probability.
There will be an intersection if , which is represented by the blue region. The area under this curve represents all the combinations of distances and angles that will cause the needle to cross a line. The area under the blue curve, which is equal to in this case, can found by evaluating the integral
Then, the area of the sample space can be found by multiplying the length of the rectangle by the height.
The probability of a hit can be calculated by taking the number of total ways a cross can occur over the total number possible outcomes (the number of trials). For needle drops, the probability is proportional to the ratio of the two areas in this case because each possible value of θ and D is equally probable. The probability of a cross is
Using Random Samples to Approximate Pi
The original goal of the Buffon's needle method, approximating pi, can be achieved by using probability to solve for pi. If a large number of trials is conducted, the proportion of times a needle intersects a line will be close to the probability of an intersection. That is, the number of line hits divided by the number of drops will equal approximately the probability of hitting the line.
So
Therefore, we can solve for pi:
Watch a Simulation
Why It's Interesting
Teaching Materials
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References
[1] http://www.maa.org/mathland/mathtrek_5_15_00.html
[2] http://mste.illinois.edu/reese/buffon/bufjava.html
[3] http://www.absoluteastronomy.com/topics/Monte_Carlo_method
[4] The Number Pi. Eymard, Lafon, and Wilson.
[5] Monte Carlo Methods Volume I: Basics. Kalos and Whitlock.
[6] Heart of Mathematics. Burger and Starbird
[7] http://math.tntech.edu/techreports/TR_2001_4.pdf
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