Monty Hall Probability Calculator

About this calculator

The Monty Hall problem, named after the host of the game show Let's Make a Deal, is a famous probability puzzle: a contestant picks one of several doors, the host — who knows what's behind each door — opens a losing door, and the contestant is offered the chance to switch. Counterintuitively, switching is almost always the better strategy: with three doors, staying wins with probability 1/3 while switching wins with probability 2/3. This calculator generalizes the puzzle to any number of doors, showing exactly how the switching advantage shrinks (but never disappears) as more doors are added.

The problem is a mainstay of probability theory and decision-theory education because it exposes how strongly conditional information — the host's deliberate, non-random choice of which door to open — changes the odds compared to naive intuition. It routinely appears in statistics courses, game theory discussions, and even in classic debates over conditional probability that once split professional mathematicians.