The Monty Hall Problem
Three doors. A car behind one, goats behind the others. You pick a door, the host opens another to show a goat, and asks if you want to switch. Two doors left, so surely it's fifty-fifty? It isn't, and almost everyone gets it wrong. Four steps: play it, watch a crowd play it, see why with a hundred doors, then watch the answer flip when the host stops knowing.
Your turn
Behind one door is a car. Behind the other two, goats. You pick a door. Then the host, who knows where the car is, opens one of the other two to show you a goat, and offers you the chance to switch.
Pick a door. Any door.
Now watch a thousand people play
Your tally is a handful of games. So send in a crowd: 500 people who always stick, and 500 who always switch. Each dot is one game. Lit means they drove home in the car.
Why: add more doors
Three doors hide the trick. Try it with a hundred. You pick one. The host, who knows where the car is, opens every other door but one, all goats. Would you still think your door is a coin toss?
The catch: it only works because the host knows
Now change one thing. The host has no idea where the car is and opens one of your two other doors at random. Sometimes that shows the car and the game is void. We keep only the games where, by luck, it showed a goat. Same doors, same goat on show.
What you see is the same. What you learn is not.
In both games you end up looking at two closed doors and one goat. The difference is how that goat got there. A host who knows had to avoid the car, so the door the host skipped is telling you something. A host who guessed just got lucky, and luck tells you nothing about your door.
Probability isn't only about what happened. It is about how you came to find out. The same evidence, chosen by someone with knowledge, means something different from the same evidence turning up by chance.
Worth knowing
A game show and a newspaper column
It is named after Monty Hall, host of the American game show Let's Make a Deal. The statistician Steve Selvin posed it in 1975, and it went famous in 1990 when Marilyn vos Savant answered it in Parade magazine: switch. Around ten thousand readers wrote in to say she was wrong, many of them with PhDs. She wasn't.
Evidence depends on how it was chosen
A goat behind door three means one thing if the host had to avoid the car and another if the host picked blindly. The fact on show is identical. What differs is the process that put it there, and odds that ignore the process come out wrong.
Someone chose what you were shown
The same trap sits under every curated sample: the reviews a seller chose to display, the case studies a vendor chose to publish, the test results a model was chosen on. Before updating on what you see, ask who picked it and what they knew.
Every game here is simulated in your browser: the car is placed at random, and the host follows the rules exactly as described in each step.
Try another puzzle →