Expected number of non-empty boxes

Question

Suppose 20 balls are thrown independently into 20 boxes (the probability that each ball lands into a particular box is 1/20). Find the expected number of non-empty boxes.

Answer 1

Hint. Let $X_i$ be the random variable which is $1$ if, at end, the $i$-th box is non-empty and $0$ otherwise.

What is $P(\{X_i=1\})$? What is $E(X_i)$?

Notice that you are looking for $E(\sum_{i=1}^{20} X_i)=\sum_{i=1}^{20}E(X_i)$.

P.S. We have that $E(X_i)=P(\{X_i=1\})=(1-(19/20)^{20})$ for $i=1,\dots,20$.

Therefore the expected number of non-empty boxes is $$20\cdot (1-(19/20)^{20})\approx 12.83028.$$