[alangpierce]

I think the second table is more clear than the first, since it shows that Monty Hall's choice depends on your choice. The main nuance to this problem is that Monty Hall is not picking randomly. There's always some unpicked door with a goat, and he'll pick that one to reveal. So if your original choice is door #1, then when deciding to stay or switch, you're really deciding whether go with door #1 or with the better of doors #2 and #3 (since the worse of the two doors has already been shown to you).

[speakeron]

That's how I got to understand it when I first heard about the problem in the early 90s. (Of course, I also wrote a simulation.)

The problem itself can be rephrased in an even simpler manner where Monty doesn't open any doors at all. Once you've picked a door, he then gives you the option of either staying with that door or changing to both of the other doors (where you get to keep the best prize behind them). Given that formulation, almost everybody would switch.