# Probability Demos

• Objective
• Level
• Prerequisites
• Platform
• Instructor's Notes
• Credits
•
Objective: To approach the concept of probability through demos and experiments by using simulations and to introduce the Monte Carlo method.

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. Animations, Java applets, Excel routines, and MATLAB routines are available for some of the demos described. See the individual demos for specific details.

Instructor's Notes: (There are currently six demos in this collection.)

This is a collection of probability demos/experiments that can be used by instructors to lay a foundation for basic probability ideas. We describe the individual demos in the collection and provide links to the demos where they are presented in detail.

Overview: There are several demos/experiments/simulations in this collection. The topics presented are representative of a wide variety of group projects and demonstrations used by instructors to enhance understanding of basic probability. The topics are appropriate for high school, general collegiate liberal arts mathematics classes, or even probability classes for math majors. Other experiments and activities are available in the references cited in the individual demos.

The common thread in our demos is the use of the Monte Carlo simulation method. Monte Carlo simulation is a technique for approximating a desired probability without performing the actual experiment, hence no physical apparatus is required. This type of mathematical model teaches students how to represent a real-world system in terms of mathematical relations. Because the technique is commonly used to solve actual problems and is conceptually easy to introduce, it provides a good introduction to probability. In fact, there is no need to try and explain the general concept a priori, the experience students get from a few experiments often lays sufficient foundation for them to see how the technique applies in a variety of situations.

Our first demo is a classic introductory experiment for estimating the area of a circle using Monte Carlo simulation. This requires a minimal mathematical background and can easily be generalized to polygonal figures. However, we point out a novel use of telephone numbers to obtain the pairs of random numbers needed in the experiment. To view this demo click on MCArea.

The second demo involves a network simulation that can be performed using 0 - 1 spinners. This demo/experiment is ideal for classes with little or no probability background. To view this demo click on MCPump. (It is well suited for a cooperative-learning format.)

The third demo is an allocation simulation. Here we have 100 cookies to which we want to randomly distribute a fixed number of chocolate chips. In one instance a manufacturer would like to know the number of cookies that, on the average, contain no chocolate chips. This experiment is suitable for a variety of mathematical levels since theoretical probabilities can be computed. To view this demo click on MCChip.

The fourth demo is also an allocation simulation, but of a different type. A scientist collects 100 droplets from a sample of a liquid containing bacteria. When the scientist studies the droplets, she found that only 1/2 of the 100 droplets collected contained bacteria. This process was repeated a number of times and it was found that on the average only 50 of the droplets were contaminated with bacteria, but the total number of bacteria from those 50 droplets varied. We want to approximate the average number of bacteria contained in the 50 droplets of this particular liquid. To view this demo click on MCBact.

The fifth demo provides a simulation of a physical phenomena. Here we indicate how to simulate a rainbow. This demo provides a nice link between mathematics and the physics of a rainbow. To view this demo click on MCRain.

The sixth demo illustrates a simulation approach to estimate probabilities for two geometrically oriented problems involving line segments and another circles within the unit circle. This demo can be used a variety of student groups depending on how much of the material is incorporated. The demo is supported by animations, Excel routines and MATLAB routines that illustrate t he geometry involved and the estimation of the probability of a success. To view this demo click on cicrsimdemo.

This collection is open ended and will be expanded when other demos of a comparable level are received. We invite users of this project to suggest additions.

Credits:  This collection of demos was organized by Dr. David R. Hill with the assistance of  Un Jung Sin, a student at Temple University. The MATLAB and Excel files that accompany these demos were written by David R. Hill.

David R. Hill
Department of Mathematics
Temple University

See the individual demos in this collection for the original sources and developers.

DRH 4/30/00   Last updated 2/5/2006

Since 3/1/2002