alt Hacker News> the logic of Monty Hall Doors does not make sense to me
For me, the core is that you have a 1 in 3 chance of getting it right on your first guess, and nothing can change that. So if you always stick with your original guess, you will win one third of the time.
No no. The thing is, the Monty Hall guy is responding to YOUR choice. So if he has to open a door where you fail, it's a response to what he knows of your choice, so HE knows what YOU chose and is not only revealing the remaining losing choice but also the winning choice. Call it a coin flip except for he always has to call tails.
Therefore your choice can either be cadillac or goat, he cannot choose cadillac and has to show a goat, so the remaining option you DIDN'T highlight is that much more likely to be cadillac because it could've been either, but he doesn't get to pick randomly, he had to show which one was NOT the winning one.
Hence the result. And since it started out as one pick of three, he responds to you and then you respond to the added information by switching and that's where the 66% odds come from: two moves each responding to each other.