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.
