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  1 and -1  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 x-coordinate as described above, repeat the procedure on the phone number in the adjacent column of the page to obtain the corresponding y-coordinate. Naturally we could use calculators or computers to generate the pairs (x, y). However, the phone book generation enhances the hands-on 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 3-dimensions (a sphere within a cube), or going further to n-dimensions (an n-ball in an n-dimensional 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 web-based simulation for this type of probability demonstration is 
          available at
          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.

DRH 4/30/01