Monty Fall: the host opens a door at random
Three doors, one car, two goats. You pick Door 1. This host does not know where the car is: he stumbles and opens one of the other two doors uniformly at random, it happens to be Door 3, and it happens to reveal a goat.
Should you switch to Door 2? What is the probability the car is behind it?
Show a hint
In the classic version the host's choice is forced when you're wrong. Here it isn't. Write out the joint probabilities of (car location, door opened) and condition on what you saw.