Two doors left, one prize. Do you switch?

Trust your gut on this one first, then watch it break.

3 quick questions · about 2 min · no sign-up

Question 1 of 3

Three doors: one car, two goats. You pick Door 1. The host, who knows every door and always opens a goat then offers the switch, opens Door 3 to show a goat. He offers Door 2. Best move?

• Doesn't matter — two doors left, so it's 50/50
• Switch to Door 2
• Stay on Door 1
• I'm not sure

You said: Doesn't matter — two doors left, so it's 50/50

You read it as a clean coin flip, which is the trap. Switching actually wins 2/3 of the time. The host didn't reveal a random door — he was forced to dodge the car, and that constraint is exactly what breaks the 50/50 symmetry.

You said: Switch to Door 2

Right — switching wins 2/3 of the time. Your first pick was locked at 1/3 before the host moved, and the leftover 2/3 had nowhere to go but the other door.

You said: Stay on Door 1

You held your door, and you're half right — it really is still 1/3, since the host's move didn't change it. But that's the catch: the leftover 2/3 doesn't vanish, it all piles onto Door 2. Switching wins 2/3 of the time.

You said: I'm not sure

No problem — switch. It wins 2/3 of the time. The key idea: your door was fixed at 1/3 before the host acted, so the remaining 2/3 collapses onto the one door he left closed.

Another way to see it

Another angle: scale it up to 100 doors, one car. You pick Door 1 — a 1-in-100 shot, almost surely a goat. The host opens 98 other doors, all goats, leaving your door and one other. Your blind guess didn't suddenly become great; the car was almost certainly in the 99 you skipped, and he just cleared every empty one but that last door. He's basically pointing at it. Three doors is the same trick, quieter.

So the magic is in that 2/3 that doesn't disappear. Let's track where it goes.

Question 2 of 3

Before the host opens anything, your Door 1 has a 1/3 chance and the other two doors together hold 2/3. After he opens a goat door, what happens to that 2/3?

• It splits evenly — both remaining doors become 1/2
• It all collapses onto the single door the host left closed
• It disappears — opening a goat removes that probability
• I'm not sure

You said: It splits evenly — both remaining doors become 1/2

You let the reveal reset everything to even, but your door can't change. It was locked at 1/3 the instant you picked, before the host moved. So the full 2/3 doesn't split — it lands entirely on the one door he left closed.

You said: It all collapses onto the single door the host left closed

Exactly. Your 1/3 was set before he acted and stays 1/3. The 2/3 that the car is 'over there' is still true — he just showed you which 'over there' door is empty, so it all funnels onto the door he left shut.

You said: It disappears — opening a goat removes that probability

Probability can't just vanish; the car is still somewhere. Your door stays locked at 1/3, so the 2/3 has to go somewhere — and it all piles onto the one closed door the host avoided.

You said: I'm not sure

Here's the key move: your door was locked at 1/3 before the host touched anything, so it can't go up. The 2/3 doesn't split or vanish — it all collapses onto the single door he chose to leave closed.

The host being forced to dodge the car is what funnels that 2/3. Let's test that with a twist.

Question 3 of 3

Suppose instead a clueless bystander randomly opened Door 3 and it happened to show a goat — host knows nothing. Now does switching to Door 2 still beat staying?

• No — now it really is 50/50, because the reveal was random, not forced
• Yes — switching always wins 2/3, a door is a door
• Yes — switching is even better now, around 90%
• I'm not sure

You said: No — now it really is 50/50, because the reveal was random, not forced

That's the insight applied. The 2/3 only collapsed onto the other door because the host was forced to dodge the car. Strip out that constraint and the reveal carries no such information — it's genuinely 50/50, and switching no longer helps.

You said: Yes — switching always wins 2/3, a door is a door

It's not the door that matters, it's how it got opened. The host's 2/3 edge came entirely from him being forced to avoid the car. A random reveal isn't constrained that way, so here it really is 50/50 — switching gains nothing.

You said: Yes — switching is even better now, around 90%

You kept the right instinct that revealed-door problems can be lopsided, but overshot. The 2/3 edge depended on the host being forced to dodge the car. A random goat-reveal removes that constraint, leaving a true 50/50 — no advantage to switching.

You said: I'm not sure

The 2/3 only piled onto the other door because the host was forced to avoid the car. A clueless bystander's reveal isn't constrained, so it carries no information — here it's a genuine 50/50, and switching doesn't help.

The takeaway

Keep this one thing: the host's edge comes from his reveal being forced, not random. Because he must dodge the car, the leftover 2/3 collapses onto the door he leaves shut — so you switch. Make the reveal random and that edge vanishes.

The pattern

Three clues, three verdicts: a streak that meant nothing, a positive test that meant little, a revealed door that meant everything. The clue never decides its own worth — the mechanism that produced it does. Before you trust or dismiss any piece of evidence, ask one thing: how was this generated? That question is the whole skill — and it's exactly the kind of move a tutor can drill into you across any subject.