A car sits behind just one of several doors, with goats behind the rest. Pick a door, and the host opens the other doors that have a goat behind them. Is it better to switch your choice, or keep it? With 3 doors it can feel confusing, but scale it up to 10 and the answer becomes far clearer.
The Monty Hall problem is a famous probability puzzle named after a real American TV game show host. It became famous for how sharply it clashes with intuition — when the answer was first published, even well-known mathematicians reportedly wrote in to protest that it had to be wrong.
Here's the key idea. When you first pick a door, the odds it's the right one are low — but after the host opens a wrong door and reveals information, the situation changes. Switching your original pick raises your odds; keeping it leaves your odds exactly where they started. The key to this whole puzzle is that the host always deliberately opens a door that isn't the right one.
With 3 doors, that difference might not feel very big. Try it with 10 doors instead, and it becomes far clearer why switching is the better choice.