The part everyone skips
The famous answer — switch, and your chance goes from 1/3 to 2/3 — depends on a condition that is usually stated so quickly that it slides past: the host knows where the car is and deliberately avoids it. That knowledge is what makes the opened door informative. Without it, an opened goat door tells you nothing you could not have worked out yourself.
Set Host to “opens at random — car shown ⇒ round is void” and run a million trials. The rounds you see are identical: you picked a door, a goat is standing behind an opened door, you are asked whether to switch. Nothing observable has changed. But switching now wins half the time instead of two thirds.
| Host behaviour | Stay | Switch |
|---|---|---|
| Knows, always opens a goat | 0.33397 | 0.66603 |
| Opens at random, void rounds discarded | 0.50098 | 0.49902 |
Those figures are from a million-trial run of the same experiment, written up in Japanese here. The simulator on this page runs the same procedure in your browser, so you can reproduce them yourself.
Why the random host is exactly a coin flip
With N doors and a host who opens k of them blindly, condition on the car not having been revealed. Both strategies then win with probability 1/(N − k) — identical, for every N and every k. The asymmetry in the classic problem is not created by doors being eliminated. It is created by who eliminated them.
The hundred-door version
Drag Doors to 100 and leave the host opening 98 of them. Staying wins 1% of the time; switching wins 99%. This is the version that convinces people who refuse to believe the three-door case, and it works because it makes the host’s knowledge impossible to ignore — to open 98 doors without ever hitting the car, he must have been steering.
Two hosts who make the question meaningless
The last two settings are the ones Monty Hall himself pointed out in a 1991 interview: the host is not obliged to offer anything. A host who only offers the switch when you have already picked the car turns switching into a guaranteed loss. A host who only offers it when you have picked a goat turns switching into a guaranteed win. The probability of the car being anywhere never changed; only the host’s policy did. That is the whole lesson of the problem, and it is why “there are two doors left so it’s 50/50” is wrong for a reason that has nothing to do with counting doors.