Why switching raises your win odds
Enter the total number of doors and how many empty doors the host opens, and this calculator returns the win probability for keeping your first pick and for switching to a remaining door. Staying wins with probability 1/N, while switching wins with (1 โ 1/N) / (N โ 1 โ k). With 3 doors and 1 door opened, staying is 1/3 and switching is 2/3, exactly twice as good; with 10 doors and 8 opened, staying is 10% and switching is 90%, nine times as good.
The result assumes three host rules: the host never opens the prize door, never opens the door you picked, and always opens an empty door. Break any one of them and 2/3 no longer holds. A host who opens a door at random, prize included, creates a different problem entirely.
This tool works from the formula only. It does not run a random simulation and does not repeat trials to compare counts against theory. Treat it as study material for probability; written as of October 2026.
Frequently asked questions
Your first pick keeps its 1/3 chance, so the remaining 2/3 collects on the one door left.
Then 2/3 no longer holds. This calculator only covers a host who opens empty doors.
Up to the total door count minus 2, because the prize door and your door are never opened.