Area Estimates
by Monte Carlo Simulation


Objective:
To
estimate the area of a circle using a probability experiment employing
the Monte Carlo technique and to indicate how our approach can be generalized
to polygonal regions. (For a general description of the Monte Carlo technique
go to the Elementary
Probability Demo Collection.)
Level: High
school or precalculus, or even probability courses for math majors.
Prerequisites:
Basic
probability concepts; knowledge of equally likely chance and the chance
of winning a lottery.
Platform: A
phone book or a calculator or computer which
has a function that generates random numbers.
Instructor's
Notes:
We start with a typical problem that can be solved
by geometry or estimated by Monte Carlo simulation experiments. We extend
the solution of this basic problem to the estimation of an area.
Find the probability p of selecting a
point on or in the interior of a circle of radius 1 which is inscribed
in a square with side 2. (Assume that the circle is centered at the origin.)
To use Monte Carlo simulation we use a (pseudo) random
number generator to obtain ordered pairs (x, y) where 1
x 1 and 1
y 1. If
then (x, y) is in the circle. When a large number of pairs (x ,y) are chosen,
then the ratio
approximates the probability p. It follows that
the area of the circle is approximately .
An interesting procedure for generating
the random numbers is to use a telephone directory. The last two, three,
or four digits of the telephone numbers in any column of any page can be
used as a coordinate of a point in the square, (The first three digits
of a telephone number represent a locality and are not random.) Using the
last four digits a decimal between 1 and 1 is encoded as follows:
If the leading digit is even,
then the decimal is taken to be positive; otherwise it is negative. The
remaining three digits are used to form a decimal of the style .
Having generated an xcoordinate as described
above, repeat the procedure on the phone number in the adjacent column
of the page to obtain the corresponding ycoordinate. Naturally we could
use calculators or computers to generate the pairs (x, y). However, the
phone book generation enhances the handson aspect and seems to stimulate
a real involvement with the overall process.
This experiment works very well using collaborative
groups. Each group selects a page from the telephone directory and two
adjacent columns. Students have enjoyed this approach and there is an opportunity
for this to evolve into team projects. Such extensions may involve areas
of polygonal figures, going to 3dimensions (a sphere within a cube), or
going further to ndimensions (an nball in an ndimensional hypercube).
The animation in Figure 1 illustrates the
probability computation described above.
Figure 1.
This demo is based on the following work: Regina Brunner, "Numbers, Please!
The Telephone Directory and Probability", The Mathematics Teacher,
Vol. 90, No. 9, Dec. 1997, pp. 704 705.
A nice webbased simulation for this type of probability demonstration
is
available at
http://www.exploremath.com/activites/activity_list.cfm?categoryID=13.
On this site students are able to see a computer simulation of the Monte
Carlo
method and can generalize the area example by changing the size of the
circle
and a surrounding rectangle.
For a more general setting for area estimation go to Monte
Carlo Demo.
Credits:
This demo was submitted by
Regina
Brunner
Assistant to the Provost for Research
and Planning
Kutztown University
and is included in Demos
with Positive Impact with her permission.
We also acknowledge contributions by Un
Jung Sin , student at Temple University.
