Use indicator random variables to compute the expected value of the sum of \(n\) dice.

Since it is not specified, I must assume the dice in question are 20-sided. Our sample space is \(S = \{1, 2, 3, \dots, 20\}\) with \(Pr\{1\} = Pr\{2\} = \dots = Pr\{20\} = \frac{1}{20}\). We define our indicator random variable \(X_i\) associated with the die showing value \(i\).

The expected value of a dice roll is then

\[\begin{split} E[X_k] &= \sum\limits_{i=1}^{20} i \cdot Pr{X_k = i} \\ &= \frac{1 + 2 + \dots + 20}{20} \\ &= \frac{230}{20} \\ &= 11.5 \\ \end{split}\]

Rolling \(n\) dice is a collection of \(n\) independent events and thus the expected sum is simply \(11.5 \cdot n\).