Count Buffon's Needle

Date: 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/