Thank you for visiting this site. This article covers the “Monty Hall Problem.”
Named after an American TV game show, this problem caused a massive public controversy when it was published — mathematicians and even Nobel Prize winners gave the wrong answer. It is one of the most famous paradoxes in the history of probability theory.
The Setup
The rules of the game are as follows.
Three doors — A, B, and C — stand before you. Behind one door is a new car (the prize); behind the other two are goats (losing). You choose one door. Let’s say you pick Door A.
Now the host, Monty Hall, steps in. Monty knows what is behind every door, so he opens one of the doors you did not choose that “definitely has a goat.” Let’s say he opens Door C to reveal a goat.
Monty then asks you: “Would you like to switch? Stay with Door A, or switch to Door B?”
Should you switch?
Most People’s Intuition
Most people think, “It doesn’t matter — there are two doors left, so the probability is 50-50.”
This is wrong.
The correct answer is “you should switch.” Switching gives you a 2/3 probability of winning; staying gives you only 1/3. Switching doubles your chances.
Why 2/3?
Here is the simplest explanation.
When you first picked a door, the probability of being right was 1/3, and the probability of being wrong was 2/3.
Monty’s act of opening a losing door does not change “the probability that your original pick was correct.” Your chosen door still has a 1/3 chance of hiding the car.
The other two doors collectively had a 2/3 chance of containing the car. Monty revealed one of those doors as a loser — so that entire 2/3 probability concentrates on the one remaining door.
That is why switching wins 2/3 of the time.
Writing Out All Possibilities
For those not yet convinced, let’s enumerate every scenario. Suppose you first chose Door A.
| Car location | Door Monty opens | Stay result | Switch result |
|---|---|---|---|
| Door A | B or C | Win | Lose |
| Door B | Door C | Lose | Win |
| Door C | Door B | Lose | Win |
Of the three equally likely scenarios, staying wins in 1 case while switching wins in 2. Switching is twice as advantageous — plain to see.
The Great Controversy
This paradox burst into public debate in 1990. When columnist Marilyn vos Savant published the correct answer — “switch” — in her magazine column, roughly 10,000 letters of protest poured in.
Among the critics were many mathematicians and PhDs. They wrote, “Your answer is wrong” and “The probability is obviously 50-50.”
Computer simulations run tens of thousands of times, however, confirmed that switching wins approximately 2/3 of the time, proving vos Savant correct.
Why Does Intuition Fail?
The main reason people get this wrong is that they overlook the fact that “Monty’s action provides new information.”
Monty follows a strict rule: he always opens a losing door. He is not opening doors at random. If your initial choice is wrong (probability 2/3), Monty has only one losing door he can open — which means Monty’s action indirectly signals that the other door is likely the winner.
If Monty had no knowledge of the doors and opened one at random — only to reveal a goat by chance — then the probability really would be 50-50. Whether or not Monty knows the contents changes the conclusion, which is another fascinating aspect of this problem.
Summary
This article covered the “Monty Hall Problem.”
A paradox that tripped up even mathematicians illustrates how fragile human probabilistic intuition can be. Even knowing the answer, many people still feel a vague sense of disbelief — and that lingering feeling is precisely what makes this problem so memorable.
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Thank you for reading. We hope to see you in the next article.
