Buffon's Needle
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|ImageDescElem=The method was first used to approximate π by Georges-Louis Leclerc, the Comte de Buffon, in 1777. Buffon was a mathematician, and he wondered about the probability that a needle would lie across a line between two wooden strips on his floor. To test his question, he apparently threw bread sticks across his shoulder and counted when they crossed a line. | |ImageDescElem=The method was first used to approximate π by Georges-Louis Leclerc, the Comte de Buffon, in 1777. Buffon was a mathematician, and he wondered about the probability that a needle would lie across a line between two wooden strips on his floor. To test his question, he apparently threw bread sticks across his shoulder and counted when they crossed a line. | ||
- | Calculating the probability of an intersection for the Buffon's Needle problem was the first solution to a problem of | + | Calculating the probability of an intersection for the Buffon's Needle problem was the first solution to a problem of <balloon title="A probability problem is ''geometric'' if the probability can be computed as the ratio of two areas or volumes."> geometric probability</balloon> The solution can be used to design a method for approximating the number π. |
- | The solution can be used to design a method for approximating the number π | + | |
Subsequent mathematicians have used this method with needles instead of bread sticks, or with computer simulations. We will show that when the distance between the lines is equal the length of the needle, an approximation of π can be calculated using the equation | Subsequent mathematicians have used this method with needles instead of bread sticks, or with computer simulations. We will show that when the distance between the lines is equal the length of the needle, an approximation of π can be calculated using the equation | ||
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To prove that the Buffon's Needle experiment will give an approximation of π, we can consider which positions of the needle will cause an intersection. 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 horizontal lines. | To prove that the Buffon's Needle experiment will give an approximation of π, we can consider which positions of the needle will cause an intersection. 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 horizontal lines. | ||
- | The variable θ is the acute angle made by the needle and an imaginary line parallel to the ones on the paper. | + | The variable θ is the acute angle made by the needle and an imaginary line parallel to the ones on the paper. Since we are considering the case where the interline distance equals l, we might as well take that common distance to be 1 unit. |
- | + | Finally, ''d'' is the distance between the center of the needle and the nearest line. Also, there is no reason why the needle is more likely to fall at a certain angle or distance, so we can consider all values of θ and d equally probable. | |
- | Finally, ''d'' is the distance between the center of the needle and the nearest line. Also, there is no reason why the needle is more likely to fall at a certain angle or distance, so we can consider all values of θ and d equally probable. | + | |
- | 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 intersect when | + | 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 length of the green arrow, ''d'', is shorter than the length of the leg opposite θ. More precisely, it will intersect when |
<math> d \leq \left( \frac{1}{2} \right) \sin(\theta). \ </math> | <math> d \leq \left( \frac{1}{2} \right) \sin(\theta). \ </math> | ||
- | See case 1, where the needle falls at a relatively small angle with respect to the lines. Because of the small angle, the center of the needle would have to fall very close to | + | See case 1, where the needle falls at a relatively small angle with respect to the lines. Because of the small angle, the center of the needle would have to fall very close to one of the horizontal lines in order to intersect it. In case 2, the needle intersects even though the center of the needle is far from both lines because the angle is so large. |
====The Probability of an Intersection==== | ====The Probability of an Intersection==== | ||
- | In order to show that the Buffon's experiment gives an approximation for π, we need to show that there is a relationship between the probability of an intersection and the value of π. If we graph the possible values of θ along the X axis and ''d'' along the Y, we have the <balloon title="In probability theory, the sample space of an experiment is the set of all possible outcomes."> sample space</balloon> for the trials. | + | In order to show that the Buffon's experiment gives an approximation for π, we need to show that there is a relationship between the probability of an intersection and the value of π. If we graph the possible values of θ along the X axis and ''d'' along the Y, we have the <balloon title="In probability theory, the sample space of an experiment is the set of all possible outcomes."> sample space</balloon> for the trials. In the diagram below, the sample space is contained by the dashed lines. |
Each point on the graph represents some combination of an angle and a distance that a needle might occupy. | Each point on the graph represents some combination of an angle and a distance that a needle might occupy. | ||
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There will be an intersection if <math> d \leq \left ( \frac{1}{2} \right ) \sin(\theta) \ </math>, 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 intersect a line. Since each of these combinations is equally likely, the probability is proportional to the area – that's what makes this a geometric probability problem. The area under the blue curve, which is equal to 1/2 in this case, can found by evaluating the integral | There will be an intersection if <math> d \leq \left ( \frac{1}{2} \right ) \sin(\theta) \ </math>, 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 intersect a line. Since each of these combinations is equally likely, the probability is proportional to the area – that's what makes this a geometric probability problem. The area under the blue curve, which is equal to 1/2 in this case, can found by evaluating the integral | ||
- | <math>\int_0^{\frac {\pi}{2}} \frac{1}{2} \sin(\theta) d\theta</math> | + | <math>\int_0^{\frac {\pi}{2}} \frac{1}{2} \sin(\theta) d\theta = \frac {1}{2}</math> |
Then, the area of the sample space can be found by multiplying the length of the rectangle by the height. | Then, the area of the sample space can be found by multiplying the length of the rectangle by the height. | ||
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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 and conducting a large number of trials 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 and conducting a large number of trials less laborious. | ||
- | π is an irrational number, which means that its value cannot be expressed exactly as a fraction a/b, where a and b are integers. As a result, π cannot be written as an exact decimal, and mathematicians have been challenged with trying to determine increasingly accurate approximations. The timeline below shows the improvements in approximating | + | π is an irrational number, which means that its value cannot be expressed exactly as a fraction a/b, where a and b are integers. As a result, π cannot be written as an exact decimal, and mathematicians have been challenged with trying to determine increasingly accurate approximations. The timeline below shows the improvements in approximating π throughout history. In the past 50 years especially, improvements in computer capability allow more decimal places to be determined. Nonetheless, better methods of approximation are still desired. |
[[Image:history of pi2.jpg|800px]] | [[Image:history of pi2.jpg|800px]] | ||
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[[Image:box1.jpg]] | [[Image:box1.jpg]] | ||
- | These results show that the Buffon's Needle method approximation is relatively tedious. Compared to other computation techniques, Buffon's method is impractical because the estimates converge towards π rather slowly. Even when a large number of needles were dropped, this experiment gave a value of | + | These results show that the Buffon's Needle method approximation is relatively tedious. Compared to other computation techniques, Buffon's method is impractical because the estimates converge towards π rather slowly. Even when a large number of needles were dropped, this experiment gave a value of π that was inaccurate in the third decimal place. |
Regardless of the impracticality of the Buffon's Needle method, the historical significance of the problem as a Monte Carlo method means that it continues to be widely recognized. | Regardless of the impracticality of the Buffon's Needle method, the historical significance of the problem as a Monte Carlo method means that it continues to be widely recognized. |
Revision as of 15:38, 14 June 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 intersects a line.
Buffon's Needle |
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Contents |
Basic Description
The method was first used to approximate π by Georges-Louis Leclerc, the Comte de Buffon, in 1777. Buffon was a mathematician, and he wondered about the probability that a needle would lie across a line between two wooden strips on his floor. To test his question, he apparently threw bread sticks across his shoulder and counted when they crossed a line.Calculating the probability of an intersection for the Buffon's Needle problem was the first solution to a problem of geometric probability The solution can be used to design a method for approximating the number π.
Subsequent mathematicians have used this method with needles instead of bread sticks, or with computer simulations. We will show that when the distance between the lines is equal the length of the needle, an approximation of π can be calculated using the equation
A More Mathematical Explanation
Will the Needle Intersect a Line?
To prove that the Buffon's Needle experiment will give an approximation of π, we can consider which positions of the needle will cause an intersection. 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 horizontal lines.
The variable θ is the acute angle made by the needle and an imaginary line parallel to the ones on the paper. Since we are considering the case where the interline distance equals l, we might as well take that common distance to be 1 unit. Finally, d is the distance between the center of the needle and the nearest line. Also, there is no reason why the needle is more likely to fall at a certain angle or distance, so we can consider all values of θ and d equally probable.
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 length of the green arrow, d, is shorter than the length of the leg opposite θ. More precisely, it will intersect when
See case 1, where the needle falls at a relatively small angle with respect to the lines. Because of the small angle, the center of the needle would have to fall very close to one of the horizontal lines in order to intersect it. In case 2, the needle intersects even though the center of the needle is far from both lines because the angle is so large.
The Probability of an Intersection
In order to show that the Buffon's experiment gives an approximation for π, we need to show that there is a relationship between the probability of an intersection and the value of π. If we graph the possible values 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.
Each point on the graph represents some combination of an angle and a distance that a needle might occupy.
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 intersect a line. Since each of these combinations is equally likely, the probability is proportional to the area – that's what makes this a geometric probability problem. The area under the blue curve, which is equal to 1/2 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 is equal to the ratio of the two areas in this case because each possible value of θ and d is equally probable. The probability of an intersection is
Thus, in order to approximate , it remains only to approximate .
Smaurer1 18:10, 14 June 2010 (UTC) Following my general desire to separate the probability aspects of this page from the approximation aspects of this page, I think this probability section should end with the conclusion that pi can be expressed in terms of the probability. Then the next section can concentrate on the approximation. The middle two equations of that section would have to be changed.
Using Random Samples to Approximate π
The original goal of the Buffon's needle method, approximating π, can be achieved by using probability to solve for π. 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.
Also, recall from above that
So
Therefore, we can solve for π:
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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