Undergraduate Probability Topics - MathOverflow

Undergraduate Probability Topics

2010-09-01T18:10:21Z

I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or less. For example, here are two very cool topics I covered last time I taught this course:

Search engines and Markov chains; AMS' description and The $25 billion eigenvector
Benford's Law and the Pareto Distribution

Thanks in advance!

Answer by Robert Bell for Undergraduate Probability Topics

2010-09-01T19:51:39Z

Here's a cool and accessible article by David Austin on percolation: http://www.ams.org/samplings/feature-column/fcarc-percolation. And if you do a quick Google search for "java percolation simulation" you will have access to quite a few nice in class demos.

Answer by Bruno Stonek for Undergraduate Probability Topics

2010-09-01T20:39:12Z

I think you might find this MO topic interesting: http://mathoverflow.net/questions/9218/probabilistic-proofs-of-analytic-facts , especially Bernstein's proof of the Weierstrass theorem.

Answer by Joseph O'Rourke for Undergraduate Probability Topics

2010-09-01T20:42:12Z

Kenneth Levasseur's paper, "How to Beat Your Kids at Their Own Game" analyzes the simple game of guessing whether the next card in a deck is red or black. He computes the expected score of correct guesses if you count carefully. There is a nice geometric flavor to his analysis. With the standard 52-card deck, the expected score is slightly over 30.

Mathematics Magazine Vol. 61, No. 5 (Dec., 1988), pp. 301-305.

Answer by Keivan Karai for Undergraduate Probability Topics

2010-09-02T09:41:40Z

One topic with a lot of "applications" is the so-called secretary problem. http://en.wikipedia.org/wiki/Secretary_problem

You can use it as an example to introduce them to the concept of stopping time. There is a lot of variations of the problem (say, replace finding the best by maximizing the expected value) that allow them to explore different aspects of the theory.

Answer by Jon Peterson for Undergraduate Probability Topics

2010-09-02T13:05:52Z

If you talk about Markov chains at some point there are a lot of cool applications to baseball. For instance using available batting statistics you can construct a team consisting of 9 Alex Rodriguez's and compute (or simulate really) how many runs such a team would score in 9 innings. You can do more detailed analysis of players as well. One place to look for more details about this (and other fun applications in sports) is the book "Mathletics" by Wayne Winston.

http://www.amazon.com/Mathletics-Gamblers-Enthusiasts-Mathematics-Basketball/dp/069113913X/ref=sr_1_1?ie=UTF8&s=books&qid=1283432632&sr=8-1

Answer by Roberto Imbuzeiro Oliveira for Undergraduate Probability Topics

2010-09-02T15:08:44Z

Here is another suggestion involving Markov chains: Example 1 in Diaconis' The Markov Chain Monte Carlo revolution. This is a very surprising application of MCMC to decoding messages exchanged between interns in California's prision system.

Answer by Joel Reyes Noche for Undergraduate Probability Topics

2011-03-04T03:23:21Z

For an application involving game theory, try Parrondo's Paradox: "Given two games, each with a higher probability of losing than winning, it is possible to construct a winning strategy by playing the games alternately."

Answer by Matthew Kahle for Undergraduate Probability Topics

2011-03-04T04:55:02Z

The Bayer and Diaconis paper, "Trailing the Dovetail Shuffle to its Lair" is a classic and Brad Mann gave a very readable exposition of it.

Many people have heard, "seven shuffles are necessary to mix up a deck of cards." But it is great for undergraduates to learn what exactly is meant mathematically by shuffle, and especially what is meant by mix up.