🎩 Ladies and Gentlemen, the Trick Explained
Almost everyone's gut says the two remaining doors are a coin flip. The gut is wrong, and it's wrong for one reason: the host is not flipping a coin. He knows where the car is, and he never opens it. That single fact leaks information into the door he leaves closed.
Act I — There Are Only Three Ways the Game Can Start
You don't need probability theory for this one, just patience. Fix your first pick as door 1 (the labels are arbitrary, so this costs nothing). The car is behind door 1, 2, or 3 — each equally likely. Play out all three and count the wins.
Count them: staying wins in exactly 1 of the three equally likely cases. Switching wins in 2. That is the whole proof.
The asymmetry comes from the host. When your first pick is wrong — which happens 2 times out of 3 — he has no choice at all: he must leave the car behind the one door he doesn't open. Switching cashes in every one of those cases.
Act II — The Same Thing, With Arithmetic
Here is every path the game can take, with the probability of each step. The one case where you picked the car splits in two (the host may open either goat), which feels like it should even things out. It doesn't — splitting 1/3 into two halves just gives you two paths worth 1/6 apiece.
Add up the gold weights on the winning leaves: staying wins on 1/6 + 1/6 = 1/3, switching wins on 1/3 + 1/3 = 2/3. Splitting the top branch in two doesn't help staying — each half is only worth 1/6.
Your first pick is right with probability 1/3, and nothing the host does afterward changes that — he was always going to open goats. So the door you're holding stays at 1/3, and the entire remaining 2/3 collapses onto the single door he left closed.
Act III — Now Do It With a Hundred Doors
If 2/3 still feels like a coin flip, crank the numbers until your intuition catches up. Pick 1 door out of 100 and watch the host — who knows exactly where the car is — throw open 98 empty ones. Ask yourself whether he did that by luck.
One hundred doors. One car. Ninety-nine goats. Pick a door — any door.
Practice round — nothing here touches your stats.
Your door was a 1-in-100 guess and it still is. The other closed door is where the other 99 chances went. The 3-door game is exactly this, just quieter.
Still Not Convinced?
Arguments are cheap. Evidence is free. Let the computer play the game a few thousand times and see which strategy the numbers pick.