The Monty Hall Problem
Three doors, with a pickup truck behind one and Elon Musk behind the other two. You pick a door. The host, who knows where the truck is, opens one of the other doors to show you Elon Musk, then asks: do you want to switch?
Play
Pick a door.
Your record
- Stayed
- —
- 0 of 0 won
- Switched
- —
- 0 of 0 won
Simulate
A few rounds by hand won't settle it. Run the game many times with a random pick each round and compare the two strategies.
Why switching wins
Your first pick is right one time in three. That never changes: nothing the host does afterward moves the truck. So two times in three the truck is behind one of the two doors you didn't pick, and the host, who must open a door with Elon Musk behind it and can't open yours, is forced to show you exactly which of those two it isn't. Switching is a bet that your first guess was wrong, and it usually was.
The trap is treating the host's reveal as fresh, neutral information: "two doors left, so it's fifty-fifty." But the host isn't opening a random door. He's constrained by knowledge you don't have, and that constraint leaks information about the door he chose not to open. If a door had been opened at random (and happened not to reveal the truck), the odds really would be even.
It helps to imagine a hundred doors. You pick one; the host opens ninety-eight Elon Musks and leaves one door closed. Do you keep your 1-in-100 guess, or take the door he conspicuously skipped?
Enumerate it
| Truck is behind | You pick 1 | Host opens | Stay | Switch |
|---|---|---|---|---|
| 1 | 1 | 2 or 3 | Win | Lose |
| 2 | 1 | 3 | Lose | Win |
| 3 | 1 | 2 | Lose | Win |
Each row is equally likely. Stay wins in one of the three; switch wins in two. The same table holds whichever door you start with.