I just had an ah-ha moment. I think I finally get this problem. It will definitely be the right choice to switch.
When you first choose, there is a 66% chance you picked the wrong door. This means there is a 66% chance that the correct door is one of the other two.
Then one of those two is eliminated. This does not change the fact that there is a 66% chance that one of those two is correct, except now you know one of them that isn't the correct door. So there is a 66% chance that the other door is the right one. You should switch.
I think the hang up for most people (certainly for me) is that it seems like the choice between two doors is independent of the original choice between three doors. What I just realized is that they aren't independent, because your first choice helps determine what doors will be available for the second choice.
When you first choose, there is a 66% chance you picked the wrong door. This means there is a 66% chance that the correct door is one of the other two.
Then one of those two is eliminated. This does not change the fact that there is a 66% chance that one of those two is correct, except now you know one of them that isn't the correct door. So there is a 66% chance that the other door is the right one. You should switch.
I think the hang up for most people (certainly for me) is that it seems like the choice between two doors is independent of the original choice between three doors. What I just realized is that they aren't independent, because your first choice helps determine what doors will be available for the second choice.