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:

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For probability and number theory, see here mathoverflow.net/questions/13011/… – Stopple Sep 1 2010 at 19:32
Some of the applications of discrete martingale theory in Williams' book are quite nice. – Steve Huntsman Sep 1 2010 at 19:41
Here is an winning application to the lottery: math.uconn.edu/~kconrad/lotterynewyork.pdf. While we're on the topic of games of chance, here is a sad lesson about negative numbers: menmedia.co.uk/manchestereveningnews/news/s/…. The second one isn't really something to discuss in class, but it should amuse the students if you just direct them to take a look at it. – KConrad Sep 1 2010 at 20:04
Yakov Sinai's book is a beautiful introduction to Probability. It covers quickly and efficiently the prerequisites and the basic material, and goes straight into more advanced topics like central limit theorems, Markov chains, branching processes, random walks, percolation. – Pietro Majer Sep 1 2010 at 20:13
Also, you might consider the equilibrium statistical physics of a 1D Ising model, or Glauber dynamics on Markov chains and simulated annealing. – Steve Huntsman Sep 2 2010 at 23:29

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.

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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.

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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.

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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.

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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.

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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.

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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."

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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.

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