Statistics Concept — Monty Hall Paradox: When Intuition Fails but Bayesian Reasoning Prevails
Last Updated on May 27, 2026 by Editorial Team
Author(s): Chao De-Yu
Originally published on Towards AI.
Statistics Concept — Monty Hall Paradox: When Intuition Fails but Bayesian Reasoning Prevails
A step-by-step probabilistic breakdown using Bayes’ theorem, graphical illustration, and Monte Carlo simulation using Python to reveal why switching improves your chance of winning
The article explains the Monty Hall Problem by framing it probabilistically: it defines events for where the prize is, how the host behaves, and what conditional probabilities mean after the host reveals an empty door. It derives the result using Bayes’ theorem (and law of total probability) to compute posterior probabilities for staying versus switching, showing that switching increases the chance of winning from 1/3 to 2/3. It then provides an intuitive graphical interpretation based on probability branches and confirms the theory with a Python Monte Carlo simulation, where the observed win rates match the predicted ~0.33 (stay) and ~0.67 (switch). Finally, the conclusion emphasizes that the host’s action is informative evidence that reshapes the probability space, making Bayesian updating—rather than intuition—the correct way to reason about the outcome.
Read the full blog for free on Medium.
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