⅔몬티홀 문제 Lab

Three doors, one car. The host just opened a goat door. Do you switch?

Switching doubles your chance of winning. The door the host opened is not luck — it is information.

Experiment

Hands-on experiment

🔮 Predict first — the host opened a goat door. Two doors remain. Is switching better?

🚪 Pick the door hiding the car

📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾

Why

Why does this exist?

In 1990, a magazine column printed the answer to this puzzle. The claim that switching doubles your chance drew over ten thousand protest letters, many from math PhDs.

Then computers played hundreds of thousands of rounds, and the column was right: 2/3 if you switch, 1/3 if you stay. It became the most famous demonstration of how badly human intuition handles probability.

The key is that the host knows. He knows where the car is and deliberately opens only goat doors. That action carries information, and information moves probability. This is the world of conditional probability.

Insight

Insights from the video

“The opened door is not luck — it is information.”
If a clueless person had opened a door and happened to reveal a goat, 50:50 would be correct. But a host who knows the answer chose that door. The moment knowledge intervenes, probability shifts.

“Short runs show luck; long runs show structure.”
In three rounds you may well lose by switching. That is why simulation matters: over 100 or 1,000 rounds the luck washes out and only the 1/3-versus-2/3 structure remains.

Misconception

Common misconceptions

Two doors remain, so it must be 50:50.

✓

Your first pick had a 1/3 chance, and the host opening a door does not change that. The remaining 2/3 simply piles onto the other door. It would only be 50:50 if the host had opened a door without knowing anything.

I tried it, switched, and lost — so switching must be wrong.

✓

Switching still loses one time in three. A probability of 2/3 means 'twice the wins in the long run', not 'always wins'. A few rounds show luck; a hundred rounds show structure.

Formula

Writing it as math

What your hands just felt, written in the language of math.

Chance your first pick is right

This does not change when the host opens a door — the pick already happened.

Chance of winning by switching

The 2/3 chance that you were wrong piles entirely onto the one remaining door, because the host removed the other goat for you.

The language of conditional probability

'Probability given this information' is called conditional probability. Monty Hall is the most famous stage on which it is born.

In Real Life

Where you meet it in real life

Ten thousand protest letters

Columnist Marilyn vos Savant published the answer and received protests from readers including a thousand PhDs. The most famous case of unanimous intuition being wrong.

Reading medical test results

'The chance you are actually ill given a positive test' depends entirely on the prior — the same information-moves-probability structure as Monty Hall.

Twenty questions, house hunting

Every question and every condition narrows the candidates. Re-computing chances whenever new information arrives is what good judgment looks like.

How AI updates beliefs

A spam filter updates 'probability of spam' with every word it reads. Monty Hall's math, matured into Bayes' theorem.

Try Yourself

Test yourself

Q1Four doors, and the host opens one goat door. If you switch to one of the two remaining doors, what is your chance?Show answer ▾

Your first pick holds 1/4. The 3/4 chance you were wrong splits over two doors: 3/8 each. Switching improves 1/4 → 3/8 — still better, just less dramatic than with three doors.

Q2One hundred doors. The host opens 98 goat doors. Should you switch?Show answer ▾

Absolutely. Your door still holds 1/100, and 99/100 sits on the single remaining door. The more doors, the more your intuition sides with the math.

Q3The host did NOT know where the car was, opened a random door, and it happened to be a goat. Is switching still 2/3?Show answer ▾

No — now it really is 50:50. A clueless opening carries no information. What moves probability is not 'a door opened' but 'someone who knows chose to open it'.

💡 Try answering yourself before revealing it — getting it wrong is where learning starts.

Watch

Concepts connect

Previous concept

P(A)

Probability

The law of large numbers — long repetition reveals structure — underpins this game.

← Probability lab

Leads to next

P(A|B)

Conditional Probability

The lab that tackles head-on why the host's action moves probability.

Go to the Conditional Probability lab →

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