Monty Hall Problem Calculator
Whether to switch doors — the counter-intuitive game-show problem, generalised to any number of doors and any number the host opens.
You pick one of n doors; the host, who knows where the prize is, opens k losing doors from the rest and offers a switch.
How the monty hall problem calculator works
You pick one of n doors; the host, who knows where the prize is, opens k losing doors from the rest and offers a switch. Your first pick was right with probability 1/n and that never changes, so all the remaining probability sits on the doors you did not pick — now concentrated on fewer of them.
Switching wins with probability (n − 1) / (n (n − k − 1)). At the classic n = 3, k = 1 that is 2/3, twice the 1/3 from staying.
Formula: P(switch wins) = (n − 1) / (n (n − k − 1))
Worked examples
| Inputs | P(win by switching) | Note |
|---|---|---|
| The classic: 3 doors, host opens 1 | 66.6667% | switching wins 2/3 |
| 100 doors, host opens 98 | 99% | switching wins 99% |
| 10 doors, host opens 5 | 22.5% | switching wins 22.5% |
FAQFrequently asked questions
Should I switch?
Yes. Switching wins two times in three; staying wins one in three.
Why isn't it 50/50 after a door opens?
Because the host is not random — he knows where the prize is and never opens it. His choice carries information, and it all flows to the door he left closed.
What if the host opened a door at random?
Then it genuinely would be 50/50 in the cases where he happens to miss the prize. The knowledge is what creates the asymmetry.
Does the 100-door version help?
It is the clearest way to see it: pick 1 of 100, the host opens 98 losers, and the one door he pointedly left closed holds 99% of the probability.
Was this actually disputed?
Famously. When Marilyn vos Savant published the correct answer in 1990, roughly a thousand PhDs wrote in to tell her she was wrong.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.