Three miniature closed wooden doors with a small concealed brass star. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
The host is not simply opening a random door
In the classic three-door problem, one door hides a prize and the initial choice is uniform. The informed host always opens a losing door that you did not choose and always offers a switch. Staying wins when the original choice was right, with probability 1/3. Switching wins when that choice was wrong, with probability 2/3.
Separate the two original cases
If your initial door has the prize, the host can reveal a loser but switching loses. If your initial door is wrong, the host's required losing-door reveal leaves the prize behind the only other closed door, so switching wins. The reveal does not turn the original choice into a fresh uniform choice between two doors.
A different host creates a different observation
If an uninformed host randomly opens an unchosen door and happens not to reveal the prize, the conditioning is different. Under the usual symmetric three-door version of that alternative mechanism, the surviving comparison can be one-half versus one-half. If a host offers switching only under selected circumstances, the offer itself may also carry information. Neither rule is the classic always-informed, always-offer protocol.
State the mechanism with the result
The Monty Hall Problem Calculator uses the classic informed-host rule and optionally simulates aggregate stay and switch results. It does not infer a host strategy from a story. Read more doors and partial reveals before extending the two-thirds answer to a larger game. The observation-pathway issue also appears in Bertrand's Box Paradox, where deliberately revealing a coin differs from drawing it uniformly.
Check the model behind your result
Use the Monty Hall Problem Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26