You're saying the chance of picking 1 door in 100 is 1/2?
No, I'm saying picking 1 door in 100 is 1/100, and picking 1 door in 2 is 1/2. That is the fundamental confusion regarding this problem.
You are given 100 choices, only one of them is correct, each one has an equal chance of being correct, and one of them must be correct. Regardless of which door you choose, you will have a 1% chance of picking the right one. This is the first stage of the problem.
Now we go through a lot of words regarding goats, sportscars, and a gameshow host. 98 choices are eliminated.
Now, how many doors are there? 2 doors. One door has a car behind it. None of the other 98 choices exist, they don't exist, and they never existed because this is a new problem. You are given a choice between two doors.
Let's start off fresh. I will give you two doors to pick from. One of them has a prize, the other does not. I guarantee you that one of the doors has a prize behind it, and that the other door does not. Now, what is the chance for each door to be the right one?
If we'd just started with this, you would have said 50%. But somehow, in a land far away, some chap herding 98 goats away from a TV studio manages to affect this probability.
Going in a completely different direction. A host comes forward with 100 doors. But before he lets you pick one, he eliminates 98 of them. Now, what is the likelihood of picking the right door?
I really don't know how else to explain this. It is not a two-part question, it is two questions. The first question asks you which of the 100 doors is correct. We never get an answer to this, and instead move on to the next question: Which of the 2 doors is correct.
I've seen the Wikipedia page, I've seen the problem outlined in Discover, and I've heard it referenced or mentioned in a number of places on the web. Not once, ever, have I seen a good explanation as to why the logic used in this problem is supposed to work.