Calculating Pi - the Nail Drop ExperimentDate: 05/19/99 at 14:43:29 From: rob Subject: Calculating pi by means of nail drop experiment I've tried to get information about "calculating pi by means of the nail drop experiment, but I've come up pretty much emptyhanded. Do you know anything about this? Thanks. Date: 05/19/99 at 17:19:36 From: Doctor Anthony Subject: Re: Calculating pi by means of nail drop experiment Needles of a length L1 are thrown at random onto a flat surface with a series of parallel lines drawn on the surface at distance L2 apart. The number of times that a needle cuts one of the lines is counted. Let the needles have length L1 and the parallel lines be drawn a distance L2 (L2 > L1) apart. A 'success' occurs when any part of a needle cuts a line. We can think of the center of the needle being uniformly distributed between 0 and L2/2. Let the smaller of the angles between the direction of a needle and the parallel lines be theta, so that theta is uniformly distributed between 0 and pi/2. If y is the distance of mid-point of the needle from the closest line, then we get an intersection if: y < (L1/2)sin(theta) We now draw two axes with y up the vertical axis varying from 0 to L2/2, and theta along the horizontal axis varying from 0 to pi/2. The sample space is any point within this rectangular area = (pi/2)(L2/2). If you draw the curve y = (L1/2)sin(theta) from 0 to pi/2, then the area under this curve divided by the total area of the rectangle will give the probability of an intersection. The area under the sine curve is INT[(L1/2)sin(theta)] = -(L1/2)cos(theta) from 0 to pi/2 = -L1/2[0 - 1] = L1/2 L1/2 Probability of an intersection = --------- (pi/2)(L2/2) 2L1 Probability = ---------- pi*L2 No.of cuts 2L1 Also Probability = ------------ = -------- No.of throws pi*L2 2L1*(Number of throws) From this pi = ----------------------- .......(1) L2*(Number of cuts) So if you throw the needle a great many times and count the number of times it cuts a line, then substitute these values into formula (1) above, you will obtain an estimate for the value of pi. You would need to repeat the process thousands of times to get a reasonably accurate estimate of pi. For more information, search the Dr. Math archives for Buffon's Needle. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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