Math

Ok, so there is a show on TV, where two guys can win a car.
There are three cars, but one key.
The first guy can choose, try the key on a car, or hand the key to the other guy so he can try.
After he used/handed the key to the other person, he/other guy can try again.
Now, three cars, one key, chance 1/3.
But, if he hands the key over, and the other guy uses it and it doesnt work, then 3/3-1/3=2/3 he has 2/3 chance?
Saw it on Mythbusters something and need to make it for homework now.
Book with the answers says 3/27, but that's just pure BS, so now I'm confused...
Also, maybe the second one has a 1/2 chance, since there are 2 cars left?