The Three Doors Problem...-s

TL;DR

This paper clarifies the distinctions among various interpretations of the classic Three Doors problem, emphasizing the importance of understanding the problem's real-world context and the diversity of mathematical solutions.

Contribution

It introduces a framework distinguishing the real-world problem from its multiple mathematical formulations and discusses three specific solutions: probability, conditional probability, and game theory.

Findings

Different solutions address different aspects of the problem.

Applied statisticians should be cautious of solution-driven approaches.

The paper highlights the importance of context in problem interpretation.

Abstract

I argue that we must distinguish between: (0) the Three-Doors-Problem Problem [sic], which is to make sense of some real world question of a real person. (1) a large number of solutions to this meta-problem, i.e., many specific Three-Doors-Problem problems, which are competing mathematizations of the meta-problem (0). Each of the solutions at level (1) can well have a number of different solutions: nice ones and ugly ones; correct ones and incorrect ones. I discuss three level (1) solutions, i.e., three different Monty Hall problems; and try to give three short correct and attractive solutions. These are: an unconditional probability question; a conditional probability question; and a game-theory question. The meta-message of the article is that applied statisticians should beware of solution-driven science.