A host who might be clumsy
Three doors, one car, two goats. You pick Door 1. With probability the host is a knowing host (always opens a goat among the other two, uniformly when he has a choice), and with probability he is a clumsy host (opens one of the other two doors uniformly at random, and might reveal the car). He opens Door 3, and it happens to show a goat.
Should you switch to Door 2? What is the probability the car is behind it?
Show a hint
Enumerate the joint outcome of (host type, car location) and keep only the branches where Door 3 is opened and shows a goat. This blends the classic Monty Hall answer with the random-fall answer.