Count Buffon's NeedleDate: 05/21/97 at 13:08:52 From: Harrieth Subject: Probability Pi For this experiment I'm doing, I need a thin stick (such as a cocktail stick or a needle) and a large piece of paper. I have to draw a series of parallel lines a distance d apart on the paper. Then I have to drop the stick at random on the paper on its end so that it "bounces". Then I have to count the number of times I drop it and the number of times it falls across one of the lines. If I drop the stick n times and it crosses a line s times, I can find the value for pi from the equation s/n = 2*l/pi*d where l is the length of the stick. Where does that equation come from? Can you give me some information about the probability involved in this experiment? Thank you very much, Harrieth Date: 05/21/97 at 17:44:38 From: Doctor Anthony Subject: Re: Probability Pi This experiment is known as Count Buffon's needle. Let theta be the acute angle the stick makes with the ruled lines and y the distance from the center of the stick to the nearest ruled line. Then the stick will cross a line if: y < (L/2)sin(theta) 0 < y < d/2 This will make the center of the stick close enough to a line for the stick to cross that line given its orientation theta. Now draw a figure with axes 0 to pi/2 along x axis, and 0 to d/2 along the vertical axis. Draw the curve (L/2)*sin(theta) between 0 and pi/2. Shade in the area below the curve. Now we get intersections if the value of theta and y put us in the shaded region. So the probability of an intersection is given by the area of the shaded region divided by the area of the rectangle with dimensions (pi/2)(d/2) = pi*d/4. The shaded region has area: INT(0 to pi/2)[(L/2*sin(theta)*d(theta)] = (L/2)[-cos(theta)] from 0 to pi/2 = -L/2 [0 - 1] = L/2 L/2 2L Required probability = -------- = ----- pi*d/4 pi*d If we carry out the experiment many times, the experimental probability should converge to this value, so we get: s 2L --- = ------ n pi*d 2L*n From which pi = ------ d*s -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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