What's So Hard about the Monty Hall Problem?

Abstract

The Monty Hall problem is notorious for its deceptive simplicity. Although today it is widely used as a provocative thought experiment to introduce Bayesian thinking to students of probability, in the not so distant past it was rejected by established mathematicians. This essay provides some historical background to the problem and explains why it is considered so counter-intuitive to many. It is argued that the main barrier to understanding the problem is the back-grounding of the concept of dependence in probability theory as it is commonly taught. To demonstrate this, a Bayesian solution is provided and augmented with a probabilistic graphical model (PGM) inspired by the work of Pearl (1988, 1998). Although the Bayesian approach produces the correct answer, without a representation of the dependency structure of events implied by the problem, the salient fact that motivates the problem's solution remains hidden.