Suppose that $10$ balls are put into $5$ boxes, with each ball independently being put in box $i$ with probability $ p_i, \sum_{i=1}^{5} p_i = 1 $
A) Find the expected number of boxes that do not have any balls.
Attempt: Let $X$ denote the number of boxes without balls. This means $ EX = \sum_{x=0}^{4} x P(X=x) $ Since we know each ball must go into at least one box, we cannot have 5 empty boxes, so that is why I sum to 4. I then said $P(X=j) = P(j{}\,\text{boxes with no balls})= {5 \choose j}(1-p_i)^{10}$ So $EX = 0 + \sum_{j=1}^{4} j{5 \choose j}(1-p_i)^{10}$ Is it ok?