Generalized Monty Hall with N Doors and K Reveals

Scenario

A game show features $N$ closed doors. Behind exactly 1 door is a car; behind the other $N - 1$ doors are goats. The contestant chooses an initial door. The host, who knows the location of the car, opens $K$ doors ($1 \le K \le N - 2$) from the unchosen doors, revealing goats. The host then offers the contestant the choice to switch to one of the remaining unopened doors.

Your Task

1. Using conditional probability and Bayes' Theorem, derive the exact win probability if the contestant stays with their original choice ($P(\text{Stay})$).
2. Derive the exact win probability if the contestant switches to a randomly selected remaining unopened door ($P(\text{Switch})$).
3. Compute both probabilities for the classic game ($N = 3, K = 1$) and an extended game ($N = 10, K = 7$).

Constraints

• Host never reveals the car and never opens the contestant's door.
• $K \le N - 2$.

Examples In classic Monty Hall ($N=3, K=1$), $P(\text{Stay}) = 1/3$ and $P(\text{Switch}) = 2/3$.