I'm usually not much a fan of these puzzle/game flavor problems, but I thought this one was really cool. Credits to the most recent BMC lecturer, Gregg Musiker.

26 students, named Aaron, Bobby, Calvin, David, Eric, ..., Zbar are playing a game. Coach Z has 26 boxes in a classroom, which contain some permutation of the student's names. One by one, the students enter the room and may open up to half the boxes, then close all the boxes and exit. The students win if and only if every student opens the box that contains his/her name during their turn.

The students may talk only before Aaron enters the room. Assuming optimal strategy, show that the probability that the students win is at least 30%.