Switching doors after a random goat reveal

On a game show there are three doors: one hides a car and two hide goats. You choose door $1$. Then an audience member, who has no idea where the car is, opens door $3$ at random and it happens to show a goat. You may now stick with door $1$ or switch to door $2$. What is the probability of winning the car if you switch?

$\tfrac12$. Unlike the classic Monty Hall setup, the reveal is not guaranteed to show a goat. Conditioning on the random door 3 showing a goat simply rules out "car behind 3", leaving doors 1 and 2 equally likely, so switching wins with probability $\tfrac12$.