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Calculating Pi - the Nail Drop Experiment


Date: 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/   
    
Associated Topics:
High School Probability
Middle School Pi
Middle School Probability

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