I'm considering rolling six fair dice. For each die, we have outcomes ${1, 2, 3, 4, 5, 6}$ that each occur with a probability of $\dfrac{1}{6}$.
The expectation, or mean value, of each die is $\dfrac{1+2+3+4+5+6}{6} = 3.5$
Now, if we take 10 of these dice, we have a range of outcomes $10, 11, 12, ..,58,59,60$.
Can we calculate the expectation of rolling the dice by calculating: $\dfrac{ \sum^{60}_{i=10} i } {50}$ ? For some reason I doubt this is right, as each outcome does not have an equal probability of occurrence.. (14 can be obtained by rolling 1,1,1,1,1,1,2,2,2,2 or 1,1,1,1,1,1,1,1,3,3 or many other rolls). Even if this is correct, it is tedious to calculate.
How can we calculate the expectation?
Thank you