Play the Monty Hall game and run thousands of simulations. Visualize why switching doors wins 2/3 of the time.
This simulator has two modes: play the classic three-door game yourself (pick a door, the host opens a goat door, then stick or switch), and run batch simulations that play thousands of trials of each strategy and plot the running win rates. The switch curve visibly settles near 66.7% while the stick curve settles near 33.3%. The thing most people get wrong is expecting 50/50 after the reveal: the host's open door is not chosen at random, and watching the curves converge makes that asymmetry visible instead of arguable.
A concrete input and expected output from the current implementation.
Input
Run 10,000 trials comparing always-switch against always-stay.
Expected output
Always switch: about 6,667 wins (~66.7%). Always stay: about 3,333 wins (~33.3%). Individual runs vary by a few percentage points; these are the values the simulator converges to.
Switching wins whenever the first pick was a goat, which happens 2/3 of the time, so its expected count is 2/3 of 10,000, about 6,667, while staying wins the remaining third, about 3,333. The plotted curves drift around these values and tighten as trials grow.
The car is placed randomly, but the reveal is not: the host must open a door that is neither your pick nor the car. When your first pick was a goat (2 games in 3), only one legal door remains, so the last unopened door hides the car and switching wins. When your first pick was the car (1 in 3), switching loses.
Each trial places the car behind a uniformly random door, draws the player's pick, performs the forced reveal, and resolves the strategy's final choice. The tool tallies wins per strategy and plots the running win rate against trial count. At 1,000 trials the switch line typically sits between 64% and 69%; at 10,000 it hovers within about a percentage point of 66.7%, which is why batch mode matters.
Because the open door is never random: it is always a goat and never your door. In the 2 out of 3 games where your first pick was a goat, the host has only one legal door left to open, so the remaining unopened door must be the car. Your door keeps its original 1/3 chance; only the other door gains probability, up to 2/3.
Then it is a plain 50/50 and the edge disappears. The 2/3 result depends on committing to a door before the reveal; with no prior pick, the host's goat-reveal leaves two doors with nothing distinguishing them, each equally likely to hide the car. The simulator plays the classic order - pick, reveal, offer - which is the order that creates the paradox.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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