system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Isn't the Monty Hall Problem two separate games?

Game One:  You are given a choice of three doors.  You pick number one.  The host opens one of the other two doors, having been given instructions that, if you pick the car, the host is to open one of the other doors, and if you pick a goat, the host opens the other door with a goat.  Stalemate.  It is a predetermined outcome.

Game Two:  The prior game's outcome stands.  The new choice you have is do you keep door number one, or do you switch?

How do you have a 2/3 chance of winning if you switch?
gpt
Because you had a 1/3 chance of picking the car on your first pick. Those odds do not change, and you don't gain new information when the host opens a goat door because you already knew one of the doors had a goat. 

The second choice is not an independent event - the chances that you are standing in front of a goat door when the "second game" starts is twice as likely as that you start in front of a car door. 



You can also just draw out all the possibilities.


Pick goat A -> Stick -> Goat A

pick goat A -> Swap -> Car

Pick goat B -> Stick -> Goat B

Pick goat B -> Swap -> Car

Pick Car -> Stick -> Car

Pick Car -> Swap -> Goat A


Two out of three times swapping will end you up with a car.