it's easier to think about it with 100 doors.
if you get to pick one and he opens 98 of the remaining ones, obviously you would switch to the remaining one you didnt pick, since 99/100 times the winning door will be in his set.
if you get to pick one and he opens 98 of the remaining ones, obviously you would switch to the remaining one you didnt pick, since 99/100 times the winning door will be in his set.
Remember, the host always knows which is the correct door, and if you selected incorrectly on the initial choice they will ALWAYS select the correct door for the second choice.
I think the easiest way to demonstrate that this is true is to play the same game with two doors, except the host doesn't open the other door if it has the prize behind it. This makes it obvious that the act of opening the door changes the probability of winning, because if the host opens the other door, you now have 100% chance of winning if you don't switch. Similarly, if they don't open the other door, you have a 0% chance of winning, and should switch. It's the fact that the host knows and chooses that is important.
It's only once you get over that initial hurdle that the 100 door game becomes "obvious". You know from the two door example that the answer isn't 50/50, and so the only answer that makes sense is that the probability mass gets concentrated in the other door.