World Wide Web resources for teaching a probability course.

Robin Lock does a good job of maintaining a website which provides information about other websites that are useful for teaching an introductory statistics course. There does not seem to be a similar website for teaching a first probability course. These remarks are meant to indicate some sources that can help in teaching a basic probability course.

Phil Pollett at the University of Queensland established the website The Probability Web to provide links to web resources for probability. This website has been taken over by Bob Dobrow at Clarkson University has made significant contributions to the site including addiing a unit on the teaching of probability.

An interactive probability book is provided by Keil Siegrist at the University of Alabama at Huntsville at his "Virtual Laboratories in Probability and Statistics". This book has a calculus prerequisite. It starts with a chapter on probability, followed by a chapter on statistics, and then a chapter on special models such as geometric probability, Poisson processes, and interacting particle systems. Siegrist has written applets to go with each topic discussed in the book.

Of course, statistics books have discussions of probability concepts and use applets to help the students understand these concepts. Particularly good examples of this are: HyperStat by David Lane at Rice University, Surfstat by John Dear at the University of Newcastle, SticiGui by T. B. Stark at U.C.Berkeley, the DAU stat refresher developed at George Mason University, "Seing Statistics" by Gary McClelland and Cyberstats by Jessica Utts and others. The latter two books are commercial products.

Applets written by Webster West and Todd Ogden for the Cyberstats project are available from Webster West's website. Individual applets that are particularly useful in teaching a probability course are: Binomial probabilities (Balasubramanian Narasimhan), Normal approximation to the binomial distribution (David Lane) Central Limit Theorem (Webster West), and Brownian motion (Charlie Geyer), Shangrong Deng at Southern State Polytech University has developed a series of applets for his Introductory Probability and Statistics course. The CUWU Statistics Program at the Univerisity of Illinois also has some interesting applets. David Harris is building a web site: Planet: probabilistic learning activities

Probability is full of surprises and the web does well by us here. Susan Holmes at Stanford University has a web site titled "Probability by Suprises". Here we find applets related to interesting probability paradoxes and problems such as: the birthday problem, the box-top problem, the hat-check problem and others. Holmes also has a detailed syllabus of her basic probability course with links to web resources used in the course. Alexander Bogomolny at his company Cut the Knot also provides a discussion of a number of puzzling math problems illustrated with Applets. His list including: Benford's law, the Buffon needle problem, Simpson's paradox, Bertrand's paradox and others. The applets of Holmes and Bogomolny have, like a fine restaurant, excellent "presentations."

An introductory probability book is available on the web."Introduction to Probability" by Charles Grinstead and Laurie Snell.

With the new software for putting video's on the web we can expect to have lectures by exceptional teachers available on the web. One example of this is the Kemeny Finite Mathematics Lectures by John G. Kemeny as Kemeny taught his last Finite Mathematics course at Dartmouth. These lectures were done before the new technology so are not technically up to current videos, but you can see a great lecturer in action. The lectures are based on the probability chapter in the first Finite Mathematics book by Kemeny, Snell,and Thompson which, thanks to Peter Doyle, is also available on the web. The Chance Lectures at the Chance website include videos of lectures and audios appropriate for a basic probability course.

Bayesian analysis is another topic that relates both to probability and statistics. The book Probability Theory: the Logic of Science by E. T. Jaynes is a very original book available on the web. Jaynes emphasizes the Bayesian and maximum entropy methods. Jim Albert at Bowling State University provides information about courses he has taught using Baysian methods at his website. Albert has also written a book Workshop Statistics: Discovery with Data, A Bayesian Approach (with Allan Rossman). Software to carry out Bayes calculations can be found from John C. Pezzullo's Web Pages that Perform Statistical Calculations. An applet illustrating test of hypothesis here. Two Bayesian web sites are: the International Society for Bayesian Analysis and the Section on Bayesian Statistical Sciences of the American Statistical Society.

There are a lot of interesting probability problems that you can find on the web. One such collection also called probability puzzles along with their solutions comes from the archives of the recreational puzzle listserve as edited for the web by Arlet Ottens. Here you will find some interesting variations on old problems such as the birthday problem. The most interesting problem we found was to show that the digits of pi are not random by exhibiting a system to make money betting on the digits that have not yet been calculated.

It is natural to use Mathematica and Maple in probability courses. Ellott Tanis at Hope College describes the uses of Maple in a probability or statistics course, particularly in discussing convolutions of distributions and the Central Limit Theorem. David Neal at Western Kentucky University has developed a series of probability projects using Mathematica.

Using computer software in teaching a probability course. Computer software has made a major change in the way probability and statistics is taught. The biggest impact has been made by the ability to easily simulate chance experiments. In statistics this has allowed important probability concepts such as the Binomial distribution, law of large numbers, and Central Limit Theorem to be understood by the students without having to spend time developing formal probability and combinatorial theory. Many of the reform statistics courses have taken advantage of this. Probability courses naturally continue to teach formal probability theory but use simulations to help the students understand the formal mathematical model for probability. Most elementary statistical packages have the ability to do simple simulations with their software. Fathom includes some standard computer language instructions that permit students to simulate more complicated experiments such as toss a coin until the first head comes up. Minitab includes procedures essentially equivalent to a programming language and so quite general simulations can be carried out using Minitab. The resampling software consists of easy to use macros which allow the students to simulate a wide variety of problems. They demonstrate this by solving all the problems in Mostellar's book: 50 Challenging Probability problems by simulation.

Of course one of the greatest strengths of the web is the ability to find very current information probability theory that cannot yet be found in textbooks but might be fun to use in a probability course. The classic example of this is David Griffeath's "Primordial Soup Kitchen". For the past ten years David has provided each week a new beautiful colored pictures showing how simple cellular automaton rules create fascinating structure from random initial states. Another good example of this is the "Web Site for Perfectly Random Sampling with Markov Chains" which provides the latest information on the use of Markov Chain simulations which have recently found numerous applications in physics, statistics, and computer science.