Brain Teasers · The Third Door
Monty Hall Puzzle - Why Switching Wins 2/3
The Monty Hall problem as a story: you pick a door, the host opens a goat, and offers a switch.
What is the Monty Hall problem?
The Monty Hall problem is named after the longtime host of the game show Let's Make a Deal, though Monty Hall himself didn't invent it. The puzzle was formally posed by the statistician Steve Selvin in a 1975 letter to The American Statistician, and it became widely known in 1990 when a reader posed it to Marilyn vos Savant's 'Ask Marilyn' column in Parade magazine. Her answer provoked an enormous public backlash, including from professional mathematicians who insisted she was wrong - she wasn't.
The setup: there are three doors, a car behind one and a goat behind each of the other two. You pick a door. The host, who knows where the car is, then opens a different door that he knows has a goat behind it, and offers you the chance to switch to the remaining unopened door. The question is whether switching improves your odds of winning the car.
The correct answer is that switching wins two-thirds of the time and staying wins only one-third, which is deeply counterintuitive to most people's first instinct that two closed doors must mean 50/50 odds. The resolution is that your initial pick was only right a third of the time, and the host's move - constrained to always reveal a goat and never the car - doesn't add new information about your original door, only about the two doors you didn't pick. When your first pick was wrong, which happens two-thirds of the time, the host is forced to leave the car behind the door he didn't open, so switching captures that entire two-thirds.
The problem is famous because the two-thirds answer survives close scrutiny while feeling wrong to almost everyone on first exposure. It also has an important variant that is 50/50: if the host opens one of the other doors at random, without knowledge of what's behind it, and happens to reveal a goat, the odds do split evenly, because the reveal was not forced. The general lesson is that you have to condition on how an observation was generated, as well as on what you observed.
How it works
Three doors, one car, two goats. You pick a door; the host, who knows where the car is, opens one of the other doors to reveal a goat - he never opens the car - and offers you the chance to switch. The car was placed uniformly at random, and when your first pick is the car the host chooses between the two goat doors at random. The story asks whether you switch and what your odds are. You lock in a guess, then step through the answer and breakdown.
How scoring works
No grading - your guess is displayed back beside the answer. The story teasers share one lifetime unlock for free users.
The key insight
Your first pick was right 1/3 of the time and nothing the host does changes that - he was always going to open a goat door. When your first pick was wrong, which is 2/3 of the time, the host's constraint forces him to leave the car shut, so switching wins exactly then. The information is in how the reveal was generated: a host opening at random who happened to show a goat would leave a 50/50.
The full step-by-step solution lives at /brain-teasers/monty-hall/solution.
A worked example
Case A, first pick is the car (1/3): switching loses. Case B, first pick is a goat (2/3): the host must open the other goat, so the remaining door is the car and switching wins. Switch - it is exactly twice as good. For intuition, run it with 100 doors and 98 opened goats.
Common mistakes
• Calling it 50/50 because two doors remain - door counts are not probabilities.
• Ignoring that the host knows and must avoid the car; the random-host variant is 50/50.
• Conditioning only on what you learned, not on how you learned it.
Why interviews test this
The interview version probes whether you can say why the host's constraint matters - stating the random-host variant and why it differs shows conditional reasoning. The written solution is at /brain-teasers/monty-hall/solution.
Play The Third Door on Outcry
The Third Door is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play The Third Door and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.