Monty Hall Random Door

This variant of the Monty Hall problem asks you to reason about a door-revealing mechanism that is no longer controlled by a knowledgeable host but by a randomly chosen audience member. The twist is that a door is revealed at random and just happens, in the observed play, to be empty. You are asked whether this should change your initial choice, and to interpret the scenario under distinct assumptions about what the audience member knows and how the game is repeated. The focus is on understanding how the same visible outcome – an empty door opened – can encode very different amounts of information depending on the underlying process.

The question leans on conditional probability, Bayesian updating, and careful conditioning on events with nontrivial selection mechanisms. It tests whether you can translate a verbal game-show description into explicit probability models, separate mutually inconsistent interpretations, and reason correctly under each one. Interviewers watch for precision about sample spaces, clarity on conditioning on "an empty door being revealed" versus "a particular rule being followed," and the ability to explain whether and why the posterior probabilities over doors change.

What it tests

The core structure of this problem class is about conditional probability and information updating: when an event occurs (like a door being revealed), the way it was selected—whether by someone with knowledge or at random—changes how much information is conveyed. If the selection mechanism is informed (the selector knows where the prize is and always reveals a non-prize door), the act of revealing is not independent of the prize's location and thus updates our beliefs, leading to an uneven probability distribution over the remaining options. If the selection is uninformed and random, the event of revealing an empty door may not provide any new information, especially if it could have happened by chance. The principle is that the probability update depends on the process that led to the observed outcome, not just the outcome itself. This is why, in repeated or structured games, the mechanism of elimination matters: it determines whether the remaining options are equally likely or not.