Monty Hall Simulator
Run thousands of trials to see why switching wins 2/3 of the time — and what happens when the host doesn't know where the prize is (a common variant that confuses even mathematicians).
Setup
Win rate by strategy
Convergence (running win rate)
Why switching wins (and when it doesn't)
In the standard Monty Hall (host always knows where the prize is and never reveals it), the probability of winning by switching is N-1 / N where N = number of doors. For 3 doors, that's 2/3. By keeping, you keep your initial 1/N chance. The switch wins because your initial pick and the host's reveal are not independent events.
In the blind host variant, the host picks a door uniformly at random to open — and might accidentally open the prize. If the prize is revealed, the trial is void. Switching after a successful blind reveal is only worth 1/2 (because both remaining doors are equally likely).
Who uses it
- Educators teaching conditional probability with a famous puzzle.
- Game-show trivia buffs settling arguments about Deal or No Deal.
- Statisticians illustrating how intuitive reasoning fails on non-independent events.
Limitations
- Monte Carlo results have ~1/√N variance. With 10,000 trials, expect ~0.5% deviation from theoretical.
- We do not model "host opens door 2 vs door 3 as a strategic choice" — only the random-equivalent case.