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
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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.
Frequently asked questions
How do I use Monty Hall Simulator step by step without mistakes?
Follow the order given in the usage section: Run N trials of the Monty Hall problem. Compare switch, keep, and random strategies. Includes the host-knowledge variant where the math actually changes. The step-by-step part matters less than the unit check that sits inside it. Confirm every unit matches the thing you are actually trying to measure before you press calculate - that single check catches more wrong answers than any other habit, and it costs about ten seconds.
What are the common mistakes when using Monty Hall Simulator?
Most common mistakes come down to four habits: entering 20 when the field expects 0.2 because the field is a percentage; leaving a field blank and accepting the default without checking what it stands for; changing one input and reading a result that was calculated before the change; and quoting the output as an exact figure when the page describes it as a reference estimate. Press calculate again after any edit, and treat the number as a reference point rather than a final answer.
How do I read the result from Monty Hall Simulator?
Read the final figure first, then come back to this section for the conditions behind it: Run N trials of the Monty Hall problem. Compare switch, keep, and random strategies. Includes the host-knowledge variant where the math actually changes. Keep two things apart: the number itself, which comes from the values you typed, and the assumptions it depends on, which decide whether that number is meaningful. The page sets out what a higher or lower result means for your case, so check that direction as well as the magnitude.
Can I act on the Monty Hall Simulator result directly?
Treat it as one reference sheet before you decide, rather than as the answer itself: 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. P ProbLab A free, no-login toolkit for real probability The output describes a magnitude under the conditions you entered. It does not include what it actually costs to act, what might change while you act, or how the other side behaves. Before you commit, compare the result against reality and note which assumptions do not hold where you are.
What inputs does Monty Hall Simulator need?
Monty Hall Simulator takes 3 inputs: Number of trials、Variant、Number of doors. They are listed in the order they feed the calculation, so an earlier value can change a later one. Do not skip a field you do not recognise - work out what it means first, because a silently guessed value produces a number that looks reasonable and is not. Before typing anything, make sure every figure uses the same unit and the same time period.
What should I check before I run Monty Hall Simulator?
Three checks pay for themselves. First, confirm the question you need answered is really inside the scope of this tool rather than a neighbouring one. Second, confirm that the figures you have are drawn from the same source and the same period - mixing monthly and annual numbers, or two different definitions of the same quantity, quietly distorts the result. Third, keep the key inputs written down, so that later you can tell which set of numbers produced a figure you are about to quote.