I'll post my reply in white so it's not a spoiler for others.

When you first pick a door, your chances of it being right are one out of three.

When Monty shows you what's behind one of the other two doors, that action occurs AFTER you already made your choice.

The chances of your choice being right do not change as a result of something that happens subsequently.

Therefore, your chances of being right, if you stick with your original choice, are still one out of three.

Since there is now only one other door available, the chances of THAT door being right must be 1 minus 1/3, or two out of three.

Therefore, you should always pick the other door, not the one you started with. Your chances of winning are twice as good if you switch than if you don't switch.

Most people's intuition gets bollixed up at step 3.