Kinesys materializes in a flash in the most famous TV studio on Veritas, during the live broadcast of the quiz show Three Doors, No Lies. The audience applauds: on Veritas, a live landing counts as an excellent entrance.
The host, who like every Veritasian cannot lie, invites Liz to play. The grand prize is a brand-new flying car, with a navigation system that, he guarantees, “always takes you where you want to go.” Liz has never wanted anything so badly.
Dord grumbles that game shows are a waste of time. M00N bounces through the air like a ping-pong ball. Liz picks a door, and the host smiles…
The Riddle
You must choose which door to open among the three offered by the host. Behind one of them is a flying car; behind the other two… a goat. Liz picks a door. The host, who knows perfectly well where the car is, says to her: “Are you really sure? You can still change your mind. In fact, let me help you,” and opens one of the doors Liz didn’t pick, revealing a goat.
Assuming Liz wants to win the car, should she switch doors, stick with her original choice, or does it make no difference?
Hint
The host doesn’t open a door at random: he knows where the car is and always opens a door with a goat. If it helps, imagine the same situation with a hundred doors, ninety-eight of which the host opens.
Solution
She should always switch doors: by switching she wins the car with probability 2/3, while sticking with her original choice wins only with probability 1/3.
The key point is that the host doesn’t open a door at random: he knows where the car is and always opens one with a goat.
Suppose Liz picked door 1. At that moment the car is behind each door with probability 1/3.
- If the car is behind door 2 (probability 1/3), the host is forced to open door 3: by switching, Liz wins.
- If the car is behind door 3 (probability 1/3), the host is forced to open door 2: by switching, Liz wins.
- If the car is behind door 1 (probability 1/3), the host opens one of the other two at random (probability 1/6 each): by switching, Liz loses.
So by switching doors she wins in two cases out of three: probability 2/3. By staying, she wins only if her first choice was right: probability 1/3.