The Monty Hall Problem

A range of solutions to the Monty Hall problem is developed, with the aim of connecting popular (informal) solutions and the formal mathematical solutions of introductory text-books. Under Riemann’s slogan of replacing calculations with ideas, we discover bridges between popular solutions and rigorous (formal) mathematical solutions by using various combinations of symmetry, symmetrization, independence, and Bayes’ rule, as well as insights from game theory. The result is a collection of intuitive and informal logical arguments which can be converted step by step into formal mathematics. The Monty Hall problem can be used simultaneously to develop probabilistic intuition and to give a deeper understanding of the paradox, not just to provide a routine exercise in computation of conditional probabilities. Simple insights from game theory can be used to prove probability inequalities. Acknowledgement: this text is the result of an evolution through texts written for three internet encyclopedias (wikipedia.org, citizendium.org, statprob.com) and each time has benefitted from the contributions of many other editors. I thank them all, for all the original ideas in this paper. That leaves only the mistakes, for which I bear responsibility.