Monty Hall simulator

The host's rule matters

One door hides the prize. After your first pick, the host knows where it is, opens every other losing door except one, and always offers a switch. With N doors, exactly two stay closed: your door and the switch door. This is the informed-host version of the puzzle.

Why does switching help?

Your first pick wins only 1/N of the time. If it was wrong, the host's reveal leaves the prize behind the other closed door. Switching therefore wins (N − 1)/N of the time: 2/3 with three doors, or 99/100 with one hundred. When your first pick is the prize, the host chooses uniformly which other door to leave closed.

The Interactivate Monty Hall activity explains the classic three-door experiment. Different reveal rules, a host who may reveal the prize, or an optional switch offer create different probability problems.

Run an experiment

Apply a door count and seed, then run 1,000 paired trials. Each trial uses one prize location and one initial pick to record what would happen under both strategies. Their win counts add to the number of trials; they are counterfactual outcomes from the same games, not two independent samples.

The playable demonstration is separate and never enters the paired experiment. Identical settings and seed reproduce the experiment. Changing settings disables old results until you apply them; doing so starts a fresh experiment. Up to 50,000 trials can be accumulated and exported with the door choices and both outcomes. The pseudorandom generator is for learning, not security or prize drawings.

For another counterintuitive probability problem, try the birthday-paradox simulator. For exact arithmetic diagrams, use the fraction visualizer.