Monty Hall: Switch or Stay

This question uses a classic three-door game-show setup to probe a candidate's grasp of conditional probability and how information changes odds. After a contestant chooses one door, the informed host reveals an empty door under specific rules, then offers the chance to switch to the remaining unopened door. The candidate is asked to decide whether to switch or stay, and to justify the decision in probabilistic terms, not intuition. Variants of this puzzle are common in entry-level quant interviews and data science screens, especially where probabilistic reasoning and comfort with counterintuitive outcomes matter.

Solving it leans on conditional probability, sample-space reasoning, and especially on understanding how the host's information and constraints affect the distribution of outcomes. A rigorous answer can be framed either via explicit enumeration of scenarios or through Bayes' theorem, but must correctly condition on the host's reveal procedure. Interviewers watch for confusion between "equally likely doors" and "equally likely scenarios," and for whether the candidate can articulate how informed actions leak information about hidden states in a structured, logically consistent way.

What it tests

In problems where an agent with extra information (like the host) reveals partial information after your initial choice, the key structure is conditional probability: the likelihoods must be updated using not just the initial randomization, but also the process and constraints governing the agent's reveal. The agent's action is not independent of the underlying state; it is filtered by what the agent knows and what is possible for them to do. This means that the probability distribution over possible worlds shifts after the reveal, often in a way that is not intuitive. The principle is that the probability you initially assigned to each outcome must be reweighted by how likely the observed action would be in each case, which is formalized by Bayes' theorem. This pattern holds whenever an informed party's constrained action leaks information about hidden variables.