The Third Door

The Monty Hall problem as a story: you pick a door, the host opens a goat, and offers a switch.

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.

Why quant 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.

The The Third Door guide covers how scoring works, the strategy that wins, a worked example and the mistakes most players make.

More Brain Teasers games

• Screwy Pirates - Five rational pirates vote on how to split 100 gold coins, and you work out the split by backward induction from a one-pirate crew up.
• Tiger and Sheep - A hundred rational tigers and one sheep on a magic island, where any tiger that eats becomes a sheep itself - solved by induction on parity.
• The Split Deal - The prisoner's dilemma played as a story: two thieves, separate rooms, and one offer each.
• Stag or Hare - The stag hunt as a story: two equilibria, and a choice that comes down to trust.
• One Offer, No Discussion - The ultimatum game as a story: one split of 100 pounds, one accept-or-reject, no negotiation.
• The Growing Pot - The centipede game as a story: a pot that doubles on every pass, and logic that says take it immediately.
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