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Topic: Buffon's Needle (Read 1854 times) |
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william wu
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Buffon's Needle
« on: Apr 2nd, 2003, 4:09am » |
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While recovering from wounds in the American Civil War, a Captain Fox threw needles at a surface ruled with parallel lines, and was thus able to experimentally infer the value of pi. True story! To find out exactly how pi plays into this scenario, solve the following problem:
A surface is ruled with parallel lines. The lines are at distance D apart from each other. Suppose we throw a needle of length L on the surface at random. What is the probability that the needle will intersect one of the lines?
Note 1: A famous problem posed and solved in 1777 by French naturalist Buffon. It has long since fascinated scientists, and marks the origin of geometrical probability -- the analysis of geometrical configurations of randomly placed objects.
Note 2: These 19th century experiments began development of the Monte Carlo method, which uses repeated simulation to approximate true statistics.
Note 3: This isn't much of a riddle ... more like just an interesting mathematical exercise. The most challenging part is setting up the problem.
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| « Last Edit: Apr 2nd, 2003, 5:43am by william wu » |
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Kitty
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Re: Buffon's Needle
« Reply #1 on: Apr 4th, 2003, 12:58am » |
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We did this experiment at school last week and the answer was pi ( well 3.15 to be precise but i suppose thats close enough). The teacher showed us some A-level maths to work it out but as The class is no where near that level, no one understood. 
The experiment does work. I know cos the teacher triedto prove it to us.
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If the human brain was simple enough for us to understand, we would still be so stupid that we couldn't understand it.~Kant (1724-1804)
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william wu
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Re: Buffon's Needle
« Reply #2 on: Apr 4th, 2003, 7:01am » |
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You might understand the following already, but I'll clarify just in case someone reads this thread and misinterprets Kitty's comment that the answer is pi.
The goal of this procedure is to estimate pi, but the answer to exactly the problem I have stated (what is the probability that the needle intersects a line) is not pi. Probabilities must lie between 0 and 1, so a number around 3.14 can't be a valid answer. However, the answer is some expression in terms of L, D, and pi. (In the experiment we know L and D.)
Answer = f(L,D,pi)
One can then compute an empirical probability for the event that the needle intersects a line, and then move everything except for pi onto one side of the equation above, so you end up with an empirical value for pi.
Kitty: That's cool that your teacher performed it. How many times did he or she throw the needle?
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| « Last Edit: Apr 4th, 2003, 7:06am by william wu » |
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william wu
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Re: Buffon's Needle
« Reply #3 on: Apr 4th, 2003, 7:13am » |
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A simpler, perhaps more understandable way of estimating pi: Get a compass and draw a circle of radius R on paper. Draw a square around that circle that exactly encloses it. Cut out the square and tape it to a wall. Get a monkey to throw a bazillion darts at that square, equally at random.$ The percentage of darts that hit the inner circle should well approximate the ratio of the circle's area to the square's area. Thus:
% of inner circle hits = pi * R2 / (2R)2
pi = (% of inner circle hits) * (2R)2 / R2
pi = (% of inner circle hits) * 4
$ for nitpickers: we assume the monkey has good enough aim to always hit the board
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| « Last Edit: Apr 4th, 2003, 7:16am by william wu » |
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Kitty
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Re: Buffon's Needle
« Reply #4 on: Apr 4th, 2003, 7:33am » |
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We did the experiment in pairs and had to dropped the needle 100 times. I must say it was one of the best lesson eva as we did not hve to use to much brain power.
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| « Last Edit: Apr 4th, 2003, 7:34am by Kitty » |
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If the human brain was simple enough for us to understand, we would still be so stupid that we couldn't understand it.~Kant (1724-1804)
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cho
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I'm going to have to nitpick your monkey (I know monkeys like it when you do that), but when you say your monkey has good enough aim to always hit the board, then you assume the monkey is aiming. If he's aiming then its not random. If you taught him to throw darts at the square, he may throw all his darts at the center of the square.
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aero_guy
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Re: Buffon's Needle
« Reply #6 on: Apr 4th, 2003, 11:47am » |
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The monkey still works. You just consider the inner hits and outer hits as a percentage of total hits, rather than total throws.
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