The Monty Hall problem starts with you choosing one of many doors that hides a single prize car while every other door hides a goat. An informed host then opens K goat doors on purpose, never touching the prize when the host knows where it is. You may keep your original door or switch to one of the remaining closed options.
Switching works because your first guess is very likely wrong: it only wins with probability
1 / N. When the host reveals goats, that large probability mass that you were wrong is
concentrated into the other unopened doors. If the host opens K goats, there are
N - 1 - K other closed doors sharing roughly (N - 1) / N of the
probability—much better odds than staying. This playground lets you experience that intuition in a
single game or verify it with large simulations, even when the host is sometimes uninformed.
Simulations reuse the same door count, host behavior, and switching rules. Results stream into the metrics card so you can compare staying versus switching at scale.