Likelihood-based solution to the Monty Hall puzzle and a related 3-prisoner paradox

TL;DR

This paper presents a likelihood-based approach to solving the Monty Hall puzzle, framing it as a statistical problem, and explores a related prisoner paradox involving strategic exchanges and decision-making.

Contribution

It introduces a likelihood-based solution to the Monty Hall puzzle, applying statistical reasoning to a single game scenario, and analyzes a novel prisoner paradox involving strategic exchanges.

Findings

Likelihood approach applies to individual game scenarios

Switching decision aligns with maximum likelihood estimation

Prisoner paradox demonstrates strategic decision advantages

Abstract

The Monty Hall puzzle has been solved and dissected in many ways, but always using probabilistic arguments, so it is considered a probability puzzle. In this paper the puzzle is set up as an orthodox statistical problem involving an unknown parameter, a probability model and an observation. This means we can compute a likelihood function, and the decision to switch corresponds to choosing the maximum likelihood solution. One advantage of the likelihood-based solution is that the reasoning applies to a single game, unaffected by the future plan of the host. I also describe an earlier version of the puzzle in terms of three prisoners: two to be executed and one released. Unlike the goats and the car, these prisoners have consciousness, so they can think about exchanging punishments. When two of them do that, however, we have a paradox, where it is advantageous for both to exchange their punishment with each other. Overall, the puzzle and the paradox are useful examples of statistical thinking, so they are excellent teaching topics.