Monty Hall with a predictable host
Three doors, one car, two goats. You pick Door 1. The host knows where the car is and always opens a goat door, but he follows a fixed habit: when he has a choice of two goat doors, he opens the lowest-numbered one. He opens Door 3.
Should you switch to Door 2? What is the probability the car is behind it?
Show a hint
Ask when the host would ever open Door 3. His habit forces his hand in some car placements. Work out which placements are consistent with "opened Door 3."