The thing you’re getting by switching is the benefit of the information provided by the person who revealed an empty door.
Before a door is open, you have a 1/3 chance of selecting correctly.
After you select a door, the host picks from the other two doors. This provides extra information you didn’t have during your initial selection. The host points to a door they know is a dud and asks for it to open. So now you’re left with the question “Did I pick the correct door on the first go? Or did the host skip the door that had the prize?” There’s a 1/3 chance you picked the right door initially and a 2/3 chance the host had to avoid the prize-door.
Maybe? I don’t think that was my issue. I think I was overthinking it and using the second “choice” as an event with separate odds.
The thing you’re getting by switching is the benefit of the information provided by the person who revealed an empty door.
Before a door is open, you have a 1/3 chance of selecting correctly.
After you select a door, the host picks from the other two doors. This provides extra information you didn’t have during your initial selection. The host points to a door they know is a dud and asks for it to open. So now you’re left with the question “Did I pick the correct door on the first go? Or did the host skip the door that had the prize?” There’s a 1/3 chance you picked the right door initially and a 2/3 chance the host had to avoid the prize-door.