Associated Topics || Dr. Math Home || Search Dr. Math

### 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

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.

Needle.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability
Middle School Pi
Middle School Probability

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search