Expected amount of different diced numbers

Question

I roll n regular, d-sided dices.

What is the expected amount of different numbers on the dices? Or equivalently: What is the expected amount of numbers that have not been diced at all?

Example: n=6, d=4:

• dice 1 shows: 4
• dice 2 shows: 2
• dice 3 shows: 1
• dice 4 shows: 1

--> The amount of different numbers on the dices is 3.

Answer 1

Let random variable $X_i$ be equal to $1$ if the number $i$ is not rolled in the $n$ tosses, and let $X_i=0$ otherwise. Let $Y=X_1+X_2+\cdots+X_d$. Then $Y$ is the number of numbers not rolled.

We have by the linearity of expectation that $$E(Y)=E(X_1)+\cdots+E(X_d).$$

Note that $E(X_i)=\Pr(X_i=1)=\left(\frac{d-1}{d}\right)^n$. Multiply by $d$ to find $E(Y)$.