Ok, I've now looked at it properly, critically, I've also drunk the koolaid, and can confirm that the theory is right: you should switch every time to improve your odds to 2/3.
Without going through the whole explanation again, the core of it is that when you pick the initial door, you know nothing about what's behind any of the doors, making it 1/3 that you'll win the car. When Monty opens the door is when critical new information comes in that changes the odds: he will
always open the door with the goat behind it since he knows where the car is. If he
didn't know where the car is, then the probability would become 1/2 like people usually think initially as sometimes he would open the door with the car behind it, but that never happens. Therefore, when Monty opens the door, the 2/3 probability is concentrated in the unopened door. In the 100 door version, it becomes easier to see, since the 99% probability is now concentrated in that one remaining unopened door. Once you understand this, the whole thing falls into place.
Here's a handy online simulator to demonstrate the game in action, proving the theory. I like the way it allows you to play with the various options, including simulation speed. I suggest setting the runs to 1000 and speed to instant for the clearest results.
Monty Hall Problem --a free graphical game and simulation to understand this probability problem.
www.mathwarehouse.com
Here's 4 Numberphile videos explaining it in various ways, which helped me to understand how this works.
In the first video, a nice lady, Lisa Goldberg, an adjunct professor in the Department of Statistics at University of California, Berkeley, explains it:
In the second video, she explains it with some confusing squiggly math. It's basically the same video with the maths bits left in:
In the third video, Brady Haran, the maker of these Numberphile videos, explains it in just 4 minutes in a clear, intuitive way:
In the fourth video, Brady explains it in just
48 seconds! It's so efficient, love this.
Finally, looking at these other videos and the simulator, I think the original video's explanation in my first post is too long at almost 20 minutes and I did notice that he repeated himself several times, which only causes confusion. It actually only takes 5 minutes to explain, a bit more with the squiggly math thrown in as a bonus.
en.wikipedia.org
