Assign a number to each of those events (having 1 as outcome, having 2 as outcome, ..., having 20 as outcome) which is the job of Random Variable. For this specific example, we can assign the same numbers. The Expected Value equation is $\mu = \sum x.p(x)$. Now you have $x = {1,2,...,20}$, and if you have $p(x)$, you can easily calculate expected value.
In this case that we have a fair die with 20 sides, probability of each side is $\frac{1}{20}$. Now, we can calculate it in this way:
$\mu = \sum x.p(x) = \sum_{x=1}^{20} x.\frac{1}{20} = \frac{1}{20} \sum_{x=1}^{20} x = \frac{1}{20} \frac{20\times21}{2} = 10.5 $
(note: $\sum_{x=1}^{n} x = \frac {n\times(n+1)}{2}$)
