Expected sum if a fair die is rolled continuously

Question

The problem statement is :

What sum can I expect in rolling a fair die n times?

How to determine the mean in a convenient way? Consider the fair die is 6 sided

Answer 1

If you've used dice a lot, you may have noticed that the pips on opposite sides always add up to $7$. Since this is the same for all three pairs of opposite sides and each pair has the same chance of being rolled and both sides of each pair have the same chance of being rolled, the expected value of a single six-sided die is $7/2$. Then by linearity of expectation the expected value of $n$ six-sided dice is $n\cdot7/2$.

Answer 2

The sum $Y$ is given by $Y=X_1+X_2+\cdots +X_n$, where $X_i$ is the number rolled on the $i$th toss. We have $E(X_i)=\dfrac{21}{6}$.

Since the expectation of a sum is the sum of the expectations, $E(Y)=\dfrac{21n}{6}$.