Extending the Monty Hall problem

What is the probability of choosing the correct door if you keep your initial door? If you switch? What if there were a different number of doors? What if there were more than three doors, and more than one door held the prize? What if there were also more opportunities to stay or switch? What then would be the probability of finding the prize given n opportunities, d doors, and p prizes? What if upon each choice of door, the host reveals k goats instead of just one? What if the host reveal d - k goats, again, with d being the number of doors, and k being some whole number. Here's yet another interesting variation. And here's a different way of stating the original Monty Hall problem.