Making the Host Rule Explicit in Monty Hall-Type Problems: A Pedagogical Note

Monty Hall is easy to state but easy to under-specify. In classroom formulations, the player's choice is usually clear, but the host's rule is often left only verbally sketched, even though different host protocols can imply different posterior probabilities. This pedagogical note presents Monty Hall-type problems as finite Bayesian models with an explicit observation mechanism σ(O | p), so that the host rule is written directly into the update. Once the protocol is fixed, posterior inference is a single finite Bayes calculation, and the standard result appears as a special case. The aim is not to propose a new theorem in probability, but to offer a compact teaching representation in which the role of the host mechanism is explicit at the point where updating is performed.