arXiv:1002.3878v2 [stat.AP] 1 Mar 2010 The Three Doors Problem...-s ∗ Richard D. Gill Mathematical Institute, University Leiden, Netherlands http://www.math.leidenuniv.nl/∼gill March 1, 2010 1 Introduction The Three Doors Problem, or Monty Hall Problem, is familiar to statisticians as a paradox in elementary probability theory often found in elementary probability texts (especially in their exercises sections). In that context it is usually meant to be solved by careful (and elementary) application of Bayes’ theorem. However, in different forms, it is much discussed and argued about and written about by psychologists, game-theorists and mathematical economists, educationalists, journalists, lay persons, blog-writers, wikipedia editors. In this article I will briefly survey the history of the problem and some of the approaches to it which have been proposed. My take-home message to you, dear reader, is that one should distinguish two levels to the problem. There is an informally stated problem which you could pose to a friend at a party; and there are many concrete versions or realizations of the problem, which are actually the result of mathematical or probabilistic or statistical modelling. This modelling often involves adding supplementary assumptions chosen to make the problem well posed in the terms of the modeller. The modeller finds those assumptions perfectly natural. His or her students are supposed to guess those assumptions from various key words (like: “indis- tinguishable”, “unknown”) strategically placed in the problem re-statement. Teaching statistics is often about teaching the students to read the teacher’s mind. Mathematical (probabilistic, statistical) modelling is, unfortunately, often solution driven rather than problem driven. * v.2 // arXiv.org:1002.3878 [stat.AP] // Submitted to Springer Lexicon of Statistics 1