Should you switch in the Monty Hall problem?

Yes. Switching wins 2/3 of the time; staying wins only 1/3.

Monty Hall problem

One prize. Three doors. Should you switch?

Pick a door. The host — who knows where the car is — opens a different door to reveal a goat, then offers you the swap.

1Pick any door to begin.

⅓

Each door holds the car with probability ⅓. That changes the moment the host opens one.

Your move

Click a door on the left to make your first pick.

Switched

—

0 / 0 won

expected 66.7%

Stayed

—

0 / 0 won

expected 33.3%

Win-rate over trials0 trials

switch stay 2/3 · 1/3

How it works

In Play by hand, pick a door, see the host reveal a goat, then switch or stay. In Simulate, run trials in bulk. The bars under the doors show each door's live probability of hiding the car — watch the host's reveal shove probability mass onto the surviving door.

Why switching wins

The key: the host's choice carries information. He never opens the car door — he always opens a goat. That asymmetry concentrates probability onto the door he leaves closed.

Before any reveal, your door has a ⅓ chance and the other two share ⅔. After the host reveals one goat, your door is still ⅓ — the ⅔ didn't evaporate, it slid entirely onto the one unopened door.

Switching wins ⅔ of the time. Staying wins ⅓.

The 100-door intuition

Picture 100 doors. You pick one (1% chance of the car). The host opens 98 goat doors, leaving yours and one other. Switch?

Obviously. Your original 1% never changed — the remaining 99% piled onto that single survivor. Switching wins 99% of the time. The three-door game is the same logic at gentler odds.

The 1990 uproar

Marilyn vos Savant published the correct answer in Parade in 1990 and drew ~10,000 letters of disagreement — many from people with PhDs. Most only came around after running the simulation themselves. Which is what you just did.

Citations

Selvin, S. (1975). "A problem in probability" (letter). The American Statistician 29(1): 67. Earliest published formulation.
vos Savant, M. (1990). "Ask Marilyn." Parade, Sept 9. The column that lit the fuse.
Morgan, J.P. et al. (1991). "Let's Make a Deal: The Player's Dilemma." The American Statistician 45(4): 284–287.
Rosenhouse, J. (2009). The Monty Hall Problem. Oxford University Press.

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