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### 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.

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/
```
Associated Topics:
High School Projects
Middle School Pi

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