Expected sum if a fair die is rolled continuously 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
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 $n\cdot7/2$.
The problem can be easily solved by using the concept of indicator variables. Let $X_i$ be the random variable which counts the number of the times $i$ comes up in $n$ rolls of dice. So, we have events $X_1, X_2, X_3, X_4, X_5$ &amp; $X_6$ representing all possible scenario (for e.g. $X_1$ counts the number of times the face $1$ came up in $n$ rolls of dice).
Note that: $$ E[X_i] = P[X_i] = \frac n{6} $$

Also let's assume that random variable $X$ correspond to the sum of $n$ dice rolls. 
Then, $E[X]$ is given by:

$$E[X] = \sum_{i=1}^{i=6} i*E[X_i] = \sum_{i=1}^{i=6} i*P[X_i] = \sum_{i=1}^{i=6} i*\frac n{6} = 3.5n $$