Total number of empty boxes

Question

If you randomly put 18 balls into 10 boxes, what is the expected number of empty boxes?

I tried the following:

Let $X_{i} = {1}$ if i-th box is empty

$X_{i} ={0}$ otherwise

Then, let $Y$ be No. of empty boxes

$Y = \sum_{i=1}^{n}X_{i}$ which is the total number of empty boxes. I am unable to think of $P(X_{i} = 1)$ to give $E(X_{i})$ and hence $E(Y)$

Thanks

Answer 1

The probability that one ball doesn't go into a given box is $9/10$, so the probability that all $18$ balls don't is $P(X_i=1)=(9/10)^{18}$.