How to Settle the Monty Hall Argument

Three doors. A prize behind one. You pick a door, the host opens a different door revealing no prize, and offers you the chance to switch. Should you?

Yes — switching wins two-thirds of the time. And you will not convince anyone of this by explaining it, because the problem is famous precisely for how strongly the wrong answer feels right. Even mathematicians got this wrong in print when it was popularised.

Why switching wins

Your first pick is correct one time in three. That’s uncontroversial.

Which means it’s wrong two times in three. And in every one of those cases, the prize is behind one of the other two doors — the host, who knows where it is, opens the empty one, and the remaining door has the prize.

So: pick wrong (2/3 of the time), switch, win. Pick right (1/3 of the time), switch, lose.

P(win | switch) = 2/3

The step that trips people is treating the two remaining doors as symmetric. They aren’t. One was chosen by you at random; the other survived a deliberate filter applied by someone who knew the answer. The host’s knowledge is what transfers information.

The version that convinces people

Explanations rarely work. Two things do.

Ten doors. You pick one. The host, who knows where the prize is, opens eight empty doors. Two remain: yours, and one other. Do you switch?

Almost everyone says yes immediately, because it’s obvious that your first pick had a 10% chance and the host’s careful avoidance of one specific door is screaming information. It’s the identical logic — just scaled until the intuition stops fighting it.

Ten thousand trials. Run the three-door game repeatedly and watch switching converge on 66.7%. Arguments end quickly in front of a counter.

The condition that makes it work

The host must know where the prize is and must never reveal it. That’s the whole mechanism.

If the host opens a door at random, the games where they happen to reveal the prize are discarded — and among the remaining games, switching and staying are genuinely equal. Same doors, same visible information, completely different answer, because the process that produced the information changed.

This is why the problem is a genuinely good lesson rather than a trick: the answer depends on how the evidence was generated, not just on what the evidence is.

Why people get stuck here

• Two doors read as 50/50. The most common intuition, and it ignores how each door got there.
• The host’s knowledge overlooked. Which is the entire mechanism.
• Small samples. A hundred games can easily look like a coin flip.
• Rejecting the result on principle. People often argue harder after seeing the simulation, not less.

Common mistakes to avoid

• Explaining it with a probability tree to someone who’s already committed to 50/50.
• Forgetting to specify that the host knows and never reveals the prize.
• Running too few trials, then pointing at noise.
• Assuming the answer transfers to superficially similar puzzles where the host doesn’t know.

How to do it with Monty Hall Simulator

The Monty Hall Simulator settles it empirically, in your browser.

1. Choose a strategy — always switch, always stay, or compare both.
2. Run ten thousand games so random noise settles.
3. Compare the win rates against 1/3 and 2/3.
4. Switch to ten doors, which is the version that changes minds.

The original Parade column and its aftermath is worth reading for how strongly people resisted the correct answer. Other probability tools are in the tools directory.

Frequently asked questions

Why isn’t it fifty-fifty?

Because the host doesn’t open a door at random — they know where the prize is and never reveal it. That knowledge transfers information to the door they leave closed.

What if the host opens a door randomly?

Then games where the prize is revealed are discarded, and among the rest switching and staying really are equal. The host’s knowledge is the entire mechanism.

Why do so many people get this wrong?

Because the intuition treats two closed doors as equivalent. They aren’t: one was picked at random, the other survived a deliberate filter.

Final thought

Don’t explain it — run it, then switch to ten doors. The simulation ends the argument and the ten-door version is what makes the reason click.

Try the free Monty Hall Simulator