Expected number of pieces of a chessboard If n squares are randomly removed from a $p \ \cdot \ q$ chessboard, what will be the expected number of pieces the chessboard is divided into? 

Can anybody please provide how can I approach the problem? There are numerous cases and when I go through case consideration it becomes extremely complex.
I am posting this as an answer to make it more visible. This is currently a Brilliant.org problem, in which the specific case $ 2 \times 500 $ is asked for. While it is an interesting problem meriting discussion, that should wait until after 8:00 PM EST 5/19 when the problem set is closed.
2 =         = 2 piece
3 = (3+1)/2 = 2 piece
4 =         = 2 piece
5 = (5+1)/2 = 3 piece
6           = 3 piece
7 = (7+1)/2 = 4 piece
8 =         = 4 piece

Series = (n+1)/2 pieces

Regards,
Yuvaraj