with POSITIVE IMPACT
Auxiliary Resources for
Segments and Circles
within a Circle
(We are not responsible for any links which are not part of the "Demos" project.
When this demo was constructed the links were active.)
1. A collection of other probability demos that use Monte Carlo simulation is available at http://mathdemos.org/mathdemos/elemprob/elemprob.html . (This demo is part of the collection.) The other problems addressed in this collection are fairly simple to state, but the simulations are more complicated to construct. They accompanying software is visually appealing and can be used at a variety of levels.
2. Another demo which focuses on area estimation using Monte Carlo simulation is available at http://mathdemos.org/mathdemos/montecarlo/monte.html . The mathematical concepts that are modeled are more sophisticated than ideas in this demo.
3. The basic ideas for the two problems discussed in this demo appeared as problems in mathematics journals.
For Problem 1, see Problem 178 (proposed by Roger L. Creech) in The Two-Year College Mathematics Journal, 1980, vol. 11 No. 5, p.336; a solution appeared in The Two-Year College Mathematics Journal, 1982, vol. 13 No. 2, p.151.
For Problem 2, see the following related material; Problem 1092 (proposed by Roger L. Creech) in The Mathematics Magazine, 1980, vol. 53 No. 1, p.49; a solution appeared in Mathematics Magazine, 1981, vol. 54 No. 2, p.87.
A resource that discusses both of these problems and includes an analytic proof of probabilities of a success appears in an article by Joseph E. Chance and Pearl W. Brazier, "Two Problems That Illustrate the Techniques of Computer Simulation", Mathematics Teacher, 1986, Vol. 79, No. 9, pp.726 - 731. The exact answer for problem 1 is (1 - 3(3)1/2)/(4p) which is about 0.58650 and the exact answer for Problem 2 is 2/3.
4. A variation of Problem 2 is to require that the two points in the unit circle also be chosen so that the distance between them is less than 1. The Excel program or MATLAB program can be easily modified to estimate the probability of a success. A good question for students is, will the probability of a success in this case be larger or smaller than that for Problem 2?
5. For a brief history of the Monte Carlo method go to http://en.wikipedia.org/wiki/Monte_Carlo_method#History
Last updated 9/15/2010 DRH