Your original choice does not become more likely just because the host opens another door.
You started with:
- Door 1: 1/3 chance
- Door 2 or Door 3 together: 2/3 chance
The host then uses his knowledge to open a door that he knows contains a goat.
That is the important part.
He isn’t randomly opening one of the two doors.
He deliberately opens a door that cannot contain the car.
What Happens If You Never Switch?
Suppose you play the game 100 times and always keep your original choice.
Your first choice is correct only about one-third of the time.
So you would expect to win approximately:
33 out of 100 games.
You would lose approximately:
67 out of 100 games.
Your original choice has only a 1-in-3 probability of being correct.
What Happens If You Always Switch?
Now imagine you use a different strategy.
Every time you choose a door, you immediately switch after the host reveals a goat.
