17 balls with 20 boxes

Question

We put 17 balls into 20 boxes randomly. What is the expected value of the empty boxes?

I defined $X$ as the number of the not empty boxes, so

$X=X_{1}+X_{2}+...+X_{20}$ where $X_{i}$ is $1$ if the $i.$ box is not empty, $0$ otherwise.

I use the additive property of expected value, so

$X=20\mathbb{E}(X_{i})$ .

$\mathbb{E}(X_{i})=p=P$(the i. box is not empty) since $X_{i}$ is an indicator random variable with expected value $p$.

How can I calculate this $p$ probability?

Am I fine so far anyway?

Answer 1

p, the probability of $X_i$ is not empty is:

$$ p = 1 - (19/20)^{17} \simeq 0.5819 $$

But that probability influences that: the $X_j$ box is empty or not.

In the case that the other 19 boxes are empty, then the probability that the box 20 is not empty is 1.

Answer 2

Choose a particular box to calculate the expected value. What is the chance the first ball goes elsewhere? You have $17$ balls that all have to go elsewhere.