You pick one of three doors. The host, who knows where the car is, opens a different door with a goat, then asks: stick or switch? Almost everyone says it makes no difference. It does. Switching wins twice as often — and you can prove it here by playing two hundred games in two seconds.
What's actually happening
The setup comes from the old game show Let's Make a Deal, and the storm it caused is half the fun. In 1990, Marilyn vos Savant answered it correctly in Parade magazine — switch, it doubles your chances — and received some ten thousand letters insisting she was wrong, around a thousand of them from people with PhDs. The greatest mathematicians weren't immune: Paul Erdős reportedly refused to accept the answer until he was shown a computer simulation. The tally counter above is exactly that simulation.
Here's the way to feel it rather than fight it. Your first pick is right one time in three — nothing the host does afterwards can reach back and improve a guess you already made. So two-thirds of the time, the car is behind one of the doors you didn't pick. Now the host does you an enormous, easily-missed favour: from those two doors, he removes a guaranteed goat. He hasn't shuffled anything; he's taken the two-thirds share and funnelled all of it onto a single door. Sticking keeps your original ⅓. Switching inherits the ⅔.
The detail everything hinges on: the host knows. He always opens a goat door, never the car, never yours. If instead a clueless host opened a random door (sometimes revealing the car and spoiling the game), the leftover odds really would be 50:50, and intuition would be right. The puzzle isn't about doors at all — it's about what information someone's deliberate behaviour leaks. That instinct, formalised, is Bayesian reasoning, and it runs spam filters and medical-test interpretation alike.
- 1Hide a coin under one of three cups while a friend looks away. They pick a cup; you (knowing where the coin is) lift an empty one from the other two, and they decide: stick or switch.
- 2Play twenty rounds with them always sticking, twenty always switching, tallying wins.
- 3Sticking lands near 7/20; switching near 13/20. Watching their own tally beat their own intuition is the moment it clicks — it never works by argument alone.