If your original choice was correct, switching makes you lose.
That happens about 1/3 of the time.
But if your original choice was wrong, the host is forced to reveal the other goat, leaving the car behind the only remaining unopened door.
That happens about 2/3 of the time.
So when you always switch, you win approximately:
67 out of 100 games.
That’s twice the winning probability of always staying.
Why Does It Feel Like 50–50?
This is what makes the Monty Hall Problem so confusing.
After one goat is revealed, you can see only two unopened doors.
It feels like:
“There are two doors, so each must have a 50% chance.”
But probability isn’t determined only by how many possibilities remain.
It also depends on how those possibilities were created.
The host knew where the car was.
He didn’t randomly open a door.
He intentionally removed a losing option while preserving information about your original choice.
