Behind one of three doors is a car. Behind the other two are goats. Pick a door, then watch the host open another door to reveal a goat. Do you stay with your pick, or switch? The answer surprises almost everyone.
Your first pick locks in a 1/3 chance of being right. The host's reveal doesn't change that — but it does concentrate the remaining 2/3 chance onto the one other unopened door, making switching the better bet.
The Monty Hall Problem is named after the host of the classic game show "Let's Make a Deal." A contestant picks one of three doors, hoping to find a car behind it. The host, who knows exactly what's behind every door, then opens one of the two remaining doors to reveal a goat — never the car, and never the door the contestant picked. The contestant is then offered a choice: stick with their original door, or switch to the other unopened one. Most people's gut instinct says it shouldn't matter — after two doors remain, surely it's a 50/50 coin flip? But that instinct is wrong, and the real math is one of the most famous "aha" moments in probability.
Here's the key: when you first pick a door, you have a 1-in-3 chance of being right and a 2-in-3 chance of being wrong. That doesn't change just because the host opens another door. What the host's move does is give you new information — since the host is guaranteed to reveal a goat and avoid the car, the entire 2-in-3 chance that you were originally wrong becomes concentrated on the single remaining door. That means switching wins two-thirds of the time, while staying only wins one-third of the time. It feels almost magical, but it's simply because the host isn't opening a door at random — the host's knowledge is doing work for you.
This puzzle became internationally famous in 1990 when it appeared in Marilyn vos Savant's magazine column. She correctly explained that switching doubles your odds, and thousands of readers — including math professors — wrote in insisting she was wrong. Computer simulations (and later, careful formal proofs) eventually confirmed she was right all along. The Monty Hall Problem is now a go-to example for teaching how new information updates probabilities, a concept called Bayesian reasoning, and for showing how easily our intuitions about chance can lead us astray.
Grab three cups and a small object like a coin. Have a friend hide the coin under one cup while you look away. Pick a cup, then have your friend (who knows where the coin is) lift up an empty cup from the other two. Decide whether to switch, and track your wins over 20 rounds. Then try always staying for 20 rounds. Compare your win rates — they should land close to 33% for staying and 67% for switching, just like the simulator above!
What is the Monty Hall Problem?
It's a probability puzzle based on a game show: you pick one of three doors, the host opens a different door with a goat, and you decide whether to stay or switch.
Should you switch or stay?
You should always switch. Switching wins about 2/3 of the time, while staying only wins about 1/3 of the time.
Why does switching double your odds?
Your first pick has a 2/3 chance of being wrong, and that chance transfers entirely onto the one remaining unopened door once the host reveals a goat.
Why is this problem so famous?
When published in 1990, thousands of readers — including mathematicians — insisted the correct answer was wrong because it feels like it should be 50/50.