[jayvanguard]

This is true but it isn't really a mathematical question then with a single correct answer.

If I was in a culture where gameshow hosts behaved differently the question would have different priors and a different answer.

What if all gameshow hosts shot contestants immediately after they won? Would you be better of switching? No, you'd be dead, so you didn't "win" the car. Completely silly example but it shows why this can't be taken seriously as a mathematical question with probabilities without injecting additional assumptions (unless of course it is stated clearly but then I bet few mathematicians would get tripped up).