The Most Controversial Probability Puzzle
In 1990, Marilyn vos Savant published the correct answer to the Monty Hall problem in Parade Magazine: you should always switch doors. Nearly 10,000 readers wrote in to disagree — including hundreds of PhD mathematicians. The problem is named after Monty Hall, the host of the TV game show Let's Make a Deal, where contestants chose between three doors.
How the Game Works
You pick one of three doors. Behind one is a car; behind the other two are goats. The host, who knows what's behind each door, opens one of the remaining doors to reveal a goat. You're then offered the chance to switch to the other unopened door. Should you switch? The simulation above runs thousands of games to show that switching wins approximately 66.7% of the time.
The Bayesian Explanation
When you pick a door, you have a 1/3 chance of choosing the car. That means there's a 2/3 chance the car is behind one of the other doors. When the host reveals a goat, they don't change these probabilities — they just concentrate the 2/3 probability onto a single door. Switching captures this 2/3 probability. This is a beautiful application of Bayes' theorem in action.
Beyond Three Doors
The generalized Monty Hall problem with N doors makes the logic even clearer. Imagine 100 doors: you pick one (1% chance of being right), then the host opens 98 doors with goats. Would you switch to the one remaining door? Of course — it has a 99% chance of hiding the car. The 3-door version follows exactly the same logic, just less dramatically.