If I have 6 pictures of people and 6 names. I have to match the names to the people and getting just one right results in a victory. I have no knowledge or clues to figure out which name belongs to which person, and I must use each name once. I know that my chances are not 1 in (6!) since that is the chance that I get all of them correct, I only need to get one correct. I know my chances are greater than 1 in 6 since I have a 1 in 6 chance at the beginning to meet a win condition instantly, and if I fail then I have more chances to meet the win condition.

Trying to work this out on my own, I know the first name has 1 in 6 chance to be correct when placed to a face. After it is placed, there are two possibilities, either it is correct (which is victory and I no longer need to worry) or it is not and I must continue. In which case I move onto the next name, which has a 1 in 6 chance the face it belongs to is already taken, and 5 in 6 it wasn't already taken. This is where I get stuck and I am not sure how to continue. 

How do I calculate this probability?