The game with many doors, by counting

You will count the cases of the game with $n$ doors exactly. The prize is behind one door chosen at random. The player always picks door $0$. The host opens all the other doors except one, never opens the prize door, and leaves one door closed besides the player's door. The checker tests $3$, $4$ and $5$ doors.

The idea. If the prize is not behind the player's door, the host must leave the prize door closed. If the prize is behind the player's door, the host leaves any one of the other doors closed, each equally likely. Switching wins when the closed door holds the prize.

Your tasks.

1. Finish closed_door_choices(prize, n). Return the list of doors that the host may leave closed besides door $0$.
2. Finish p_stay(n). Add $\dfrac{1}{n}$ for each position of the prize that equals the player's door.
3. Finish p_switch(n). For each position of the prize, add $\dfrac{1}{n}$ times the fraction of the host's choices that equal the prize.