I made the common mistake that many people (including some top mathematicians) made. As it turns out, you have to set up this problem somewhat differently than you would most probability problems of a similar nature. The problem generalizes to more than three doors, but the solution (for some particular number of doors and choices of sticking with or switching a choice) isn't obvious.
There were an interesting discussion about the problem in which a poster named Steve Weimar speculated on the nature of problem solving. He noted that people develop "agents" that enable them to solve certain types of problems. But sometimes, the "agent" doesn't apply, so it's necessary to develop new "agents," and also to determine when a particular "agent" applies.
This is what I'm thinking about mostly these days - how to improve at problem solving; how to notice early on in the process what is likely to be the best approach to solving a problem. Also, if one is stuck, how to quickly backtrack to a better approach. It's like the other day, when I was stuck doing one of the aha! Insight logic problems, I couldn't figure out how to get past my mental block. But later on in that section, there was another (presumably) harder problem that I was able to solve right off the bat using some boolean algebra. Reading the problem immediately invoked that "agent" which was able to rapidly provide me with the answer, but for some reason, I didn't have an "agent" for the (presumably) easier problem.
Steve also ponders what might be the ideal environment that enables people to shift their thinking. Answers to this question might shed some light on how to improve mathematics education in the US, especially among very young children.