Difficulty: Medium · Puzzle 4 of 6 · Topic: Probability Puzzles
In a 3-door game, one car and two goats. You pick one door. The host knows where the car is and always opens a different door that hides a goat (choosing at random if he has two goat doors to choose from), then offers you a switch. Should you switch, and what are the win probabilities?
Initial pick keeps its original probability.
1) Initial chosen door has car with probability 1/3.
2) Therefore car is in one of the other two doors with probability 2/3.
3) Host always opens a goat door among those two, never opening the car door.
4) Host action transfers the full 2/3 mass to the single unopened alternate door.
5) Staying wins with 1/3; switching wins with 2/3.Answer: Switching doubles win probability from 1/3 to 2/3.
Checked by running a model of the rule in a Python model. A computational check: exact enumeration of car position and host choice gives stay 1/3 and switch 2/3 for a host who avoids the car, and 1/2 each (with a 1/3 chance of the car being revealed) for a random host.
Counting the two closed doors as 50-50. The host's choice is not random: he must open a goat door, so his action carries information and shifts the whole 2/3 onto the other door.
What changes if host sometimes opens a random door without knowing? Model answer: if the host opens a random door and it happens to show a goat, switching and staying are both 1/2. The 2/3 only appears when the host always avoids the car on purpose. (If he opens the car by accident the game just ends, with probability 1/3.)
My first pick is the car with probability one third, so the car is behind one of the other two doors with probability two thirds. The host knows where the car is and always opens a goat door among those two, so he never reveals the car. That means the whole two thirds lands on the one remaining closed door. So I should switch, and switching wins with probability two thirds while staying wins one third. The trap is calling the last two doors fifty-fifty, when the host's choice is forced and carries information.