Empty Boxes
If you randomly place 18 balls into 10 boxes (each ball independently goes into one of the 10 boxes with equal probability), what is the expected number of empty boxes?
Answer
Let \(X\) be the total number of empty boxes. Define the indicator random variable \(I_i\) for box \(i = 1,\dots,10\) by
\[
I_i =
\begin{cases}
1, & \text{if box $i$ is empty},\\
0, & \text{otherwise}.
\end{cases}
\]
Then
\[
X = \sum_{i=1}^{10} I_i.
\]
Using linearity of expectation:
\[
\mathbb{E}[X] = \sum_{i=1}^{10} \mathbb{E}[I_i].
\]
Since each ball has a \(\frac{9}{10}\) chance of not going into box \(i\), the probability that box \(i\) is empty (all 18 balls go elsewhere) is
\[
\left(\frac{9}{10}\right)^{18}.
\]
Hence,
\[
\mathbb{E}[I_i] = P(\text{box $i$ is empty}) = \left(\frac{9}{10}\right)^{18}.
\]
Therefore,
\[
\mathbb{E}[X] = \sum_{i=1}^{10} \left(\frac{9}{10}\right)^{18}
= 10 \left(\frac{9}{10}\right)^{18}.
\]
Notes and comments
Comment 1: See that this problem is very similar to one we did with dices (expected number of distinct faces).