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Monty Hall problem

Covers The Monty Hall Problem Setup, Why Switching Wins, Conditional Probability in Monty Hall and more.

Lessons4 lessons
Media
Quiz
Final assessmentAfter the last lesson, for your diploma
About

“Monty Hall problem” is a 4-lesson course at Teen level, created with soclever, a personal AI teacher. Each lesson takes a few minutes and ends with a check-in question; finish the curriculum and you can take a certificate test to earn a diploma. Try the first lesson without an account — or make your own course on any topic.

What you'll learn

  1. The Monty Hall Problem Setup. Most people think that when you pick a door at the start of this game, your odds are one in three—and that's correct.
  2. Why Switching Wins. Most people feel certain that switching doors after the host opens one shouldn't matter — that staying or switching are equally likely to win.
  3. Conditional Probability in Monty Hall. You pick a door in a three-door game. The prize is behind one door, and two doors hide nothing. Your odds of being right are one in three—that's straightforward.
  4. Real World: Monty Hall in Everyday Life. Let's start with a familiar instinct: when someone who knows more than you removes a bad option, it feels like the choices that remain are equally good.

Lesson 1: The Monty Hall Problem Setup

Most people think that when you pick a door at the start of this game, your odds are one in three—and that's correct. But here's where almost everyone goes wrong: they think those odds stay the same after new information arrives. They don't. That's the whole puzzle.

Here's the real setup. You're on a game show facing three doors. Behind one is a prize—say, a car. Behind the other two are duds, like goats. You point to one door. You don't open it yet. Your chance of having picked the car right now is one in three. The other two doors, combined, hold a two in three chance of hiding the car.

Now Monty Hall, the host, opens one of the doors you didn't pick. He always opens a door with a dud behind it—he knows where everything is, so he'll never accidentally reveal the car. One door swings open. A goat. Now two doors remain closed: the one you picked, and one other.

Here's the trap: it feels like the odds are now fifty-fifty. Two doors left, right? One must be the car. But that's wrong. Your original door still has a one in three shot. The other unopened door now holds a two in three shot. The probabilities didn't split evenly—they shifted.

This is the moment where almost everyone's intuition breaks down. The lesson to come will show you why this happens and how to see it clearly.

Questions this course answers

  • Before Monty opens any door, what is the probability that the car is behind one of the two doors you didn't pick?
  • Why does the host opening an empty door change the odds in favor of switching, when your original choice was already made?
  • Why does the unopened door the host left standing have better odds than your original choice?
  • If a knowledgeable friend who must always rule out a bad option does so after you make your first pick among three, why does switching to the remaining option give you better odds than staying?

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