Pick Your Grade

[Written by Sam Rebelsky. Based loosley on a problem from Halmos p. 58.]

Professor Randomsky has decided to use a somewhat strange process for grading his students. He has taken a hundred balls and written the letter A and fifty balls and B on fifty balls. He then distributes the balls between two boxes.

To determine the grade for a student, Professor Randomsky allows each student to choose a box and then blindly pick a ball from the box. Students are not allowed to see what's in each box. Whatever grade is on the ball is the grade the student receives.

Is there a way that Professor Randomsky can arrange the balls so that a student is more likely to get an A than a B?

How does your answer change if there are more than two boxes or more than three grades?