A row of miniature doors with several open empty frames. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Define what switching means when several doors remain
With n doors and an informed host opening m unchosen losing doors, there are n−m−1 unchosen closed alternatives. This version selects one of those alternatives uniformly when switching. Staying wins with probability 1/n. Switching wins with probability (n−1)/[n(n−m−1)]. The denominator records how the initial wrong-choice probability is spread across the remaining alternatives.
Compare partial and nearly complete reveals
For ten doors and five opened losers, four unchosen alternatives remain. Uniform switching wins with probability 9/40 = 22.5%, while staying wins with probability 10%. If the host instead opens eight losers, only one alternative remains and switching wins with probability 9/10 = 90%. Both results use the same initial prize chance; the amount of elimination differs.
Check the no-reveal boundary
When m = 0, the switching probability reduces to (n−1)/[n(n−1)] = 1/n. Merely replacing one uniform initial choice with another unchosen door adds no advantage before informed elimination. When m = n−2, switching wins whenever the initial choice was wrong, giving (n−1)/n.
Keep the host policy and switch policy explicit
The Monty Hall calculator models a host who knows the prize, never opens it and always offers switching. A player using extra information to favor one remaining door needs another switching model. The optional simulation generates observations under the same stated aggregate probabilities, not evidence about a real host. Read why host rules matter for the three-door distinction. These events are also an application of the joint probability product: initial wrong choice followed by a successful alternative selection.
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