CLASS DESCRIPTION:
Mathematical Explorations: Probability, the Improbable and the Counterintuitive
Mathematics courses often teach students how to solve problems, use algorithms, and number crunch. Even probability classes tend to focus on algoriths for solving problems in probability, rather than exploring how far one can go, and how counterintuitive some of the answers may be.
Our Special Topics and Mathematical Explorations courses teach students how to pose problems, develop algorithms, explore ideas, prove (both formally and informally) their methods and ideas work, and propose next steps. Students can use the skills learned in these classes to stretch their regular math curriculum, challenge their assumptions about mathematics, and truly think like a mathematician.
In Probability, the Improbable and the Counterintuitive, we’ll delve deeply into some astonishing ideas in probability. We’ll look at some counterintuitive problems, discussing what makes them counterintuitive and how to make them clear. We’ll pose all sorts of problems that will make us think much deeper than any basic probability problem. We’ll delve deeply into Bayesian probability, game theory, the math behind card counting and the MIT teams that took Vegas for millions, and so much more.
SYLLABUS
Week 1: Introduction to Problem Posing in Probability - coin flipping part 1
Week 2: Coin flipping part 2 - counterintuitive results
Week 3: Guessing numbers - strategies and extensions
Week 4: Using the complement - the Birthday Problem
Week 5: Monty Hall problem with extensions
Week 6: Ellsberg Paradox and other classical counterintuitive probability problems
Week 7: Introduction to expected value with some game analysis
Week 8: Card counting, MIT blackjack teams, strategy and expected value
Week 9: Game Theory Introduction, classical problems
Week 10: Bayesian Probability
Week 11: Plane problem, hat problem
Week 12: Probability and combinatorics – extended problem solving
Week 13: Game analysis
Week 14: A flexible day to catch up on anything we’ve missed or would like to extend:
Week 15: Student posed problems
(Term is 16 weeks; one of those weeks is a Fall Break)