The Generalized Monty Hall Problem

The Monty Hall Problem

The Monty Hall problem is a counter-intuitive statistics puzzle based on the game show Let’s Make a Deal.

The Setup

Suppose you’re on a game show with \(n\) doors: 1. Behind one door is a car; behind the others, goats. 2. You pick a door. 3. The host (Monty Hall), who knows what’s behind the doors, opens \(n-2\) other doors, all of which reveal goats. 4. Only two doors remain closed: your initial pick and one other door. 5. Monty asks: “Do you want to switch your choice?”

ImportantThe Paradox

With more doors, the advantage of switching becomes even more extreme. If there are 100 doors, your first pick has a 1/100 chance of being the car. Monty then filters out 98 “wrong” doors, leaving the car almost certainly behind the only other door!

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Interactive Simulator

Adjust the settings below to see how the win rate changes as you increase the number of doors (\(n\)).

viewof num_doors = Inputs.range([3, 100], {step: 1, label: "Number of Doors (n)", value: 3})
viewof trials = Inputs.range([10, 5000], {step: 10, label: "Number of Trials", value: 1000})
viewof strategy = Inputs.radio(["Always Switch", "Always Stay"], {label: "Strategy", value: "Always Switch"})

TipThe Math

Theoretical Win Rate: - Stay: \(1/n\) - Switch: \((n-1)/n\)