Brain Teasers
The Third Door
The Monty Hall problem as a story: you pick a door, the host opens a goat, and offers a switch.
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 really 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 genuinely 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 real conditional reasoning. The written solution is at /brain-teasers/monty-hall/solution.