Topics Numbers and logic

What is the Monty Hall problem?

The Monty Hall problem is a probability puzzle based on an old TV game show. You face three doors: behind one is a car, behind the other two are goats. You pick a door. The host, who knows where the car is, opens one of the other doors to reveal a goat, then offers you a chance to switch. The answer is that you should switch, because switching wins the car two times out of three, while staying wins only one time in three.

Almost everyone's first instinct is that two doors remain, so it must be fifty-fifty. That instinct is wrong, and it is why the puzzle became famous. Mathematicians and many others argued about it publicly, and plenty of people refused to believe the answer until they ran simulations. The trick is that the host is not opening a door at random. He always avoids the car, and that changes what his move tells you.

An episode on bre would walk through the puzzle slowly, test it with a simple count of every possible game, and then try a version with a hundred doors, where the logic becomes obvious. bre's hosts are AI, so they can slip, but they will show the reasoning so you can check it yourself. You can also press Talk and ask your own question.

What a bre episode would cover

An outline of the episode bre would make for this question. Every episode is written fresh when you ask, so yours will differ.

  1. The setup, one door at a timeThe rules of the game: three doors, one car, a host who knows the answer and always opens a goat door.
  2. Why fifty-fifty feels so rightWhy two closed doors seems to mean even odds, and what that intuition quietly ignores.
  3. Counting every possible gameList the three places the car could be and see what happens when you stay or switch in each case.
  4. The hundred-door versionScale the puzzle up so the host's choice stands out, and the odds stop feeling strange.
  5. What if the host did not know?How the answer changes when the host opens a door by chance, and why his knowledge is the whole point.
  6. The argument that followedHow the puzzle caused a public dispute among mathematicians and readers, and how simulations helped settle it.

How the episode might open

A sample exchange between two of bre’s AI hosts, bre and Arlo. Both are AI; this is written by AI, as every bre episode is.

  1. breAI host

    Okay, picture three doors. A car behind one, goats behind the other two. You pick door one. Then the host opens door three, and there's a goat. Do you switch?

  2. ArloAI host

    No. Two doors left. Fifty-fifty. Stay.

  3. breAI host

    That's what nearly everyone says, including a lot of people who do math for a living. And it's wrong. Switching wins about two times out of three.

  4. ArloAI host

    Who counted that?

  5. breAI host

    Fair. People ran it by hand and then by computer, thousands of games. But let's do the counting out loud, because it's quick. When you first picked, how likely were you to be right?

  6. ArloAI host

    One in three.

  7. breAI host

    Right. And opening a goat door doesn't change that, because the host was never going to open the car. So your door is still one in three.

  8. ArloAI host

    So the other door gets the leftover two in three. Huh. Fine. Walk me through it again.

Questions people also ask

Should you switch in the Monty Hall problem?
Yes, under the standard rules. Your first pick is right one time in three, and the host always reveals a goat, so the other closed door holds the car the remaining two times in three. Switching therefore wins about twice as often as staying.
Why does the Monty Hall problem feel wrong?
Two doors remain, so it seems like even odds. But the host's choice is not random: he knows where the car is and never opens it. That knowledge carries information, and it shifts the probability onto the door you did not pick.
Is the Monty Hall problem based on a real show?
Yes. It is named after Monty Hall, the longtime host of the American game show Let's Make a Deal. The puzzle is a simplified math version of that kind of door choice, and the show itself did not always follow these exact rules.
Does the answer change if the host does not know where the car is?
It can. If the host opens a door at random and happens to show a goat, switching and staying become equally likely to win. The two-thirds advantage depends on the host knowing and deliberately avoiding the car.

Related topics

More: all 300 topics, numbers and logic, or the longer reads on /learn.

bre’s hosts are AI, and every episode is generated, so they can be wrong: check anything that matters. This page outlines what an episode would cover. It is for interest and learning, not medical, financial or legal advice.