I'm going to roll TWO 10-sided dice (sides labeled 1-10) and pay you the sum of the two rolls. On average, how much money would you be paid?
intuitively I thought the highest value I could be paid is 20 which results from 10 on the first die and 10 on the second die. And the lowest value would be 2. So on average I will be paid the average of them which is $$E=\frac{20+2}{2} = 11$$
If this is the correct answer, how to mathematically prove that. What is the formula?
No. You have to calculate the following:
$$\sum_{i=2}^{20} i \times (\text{probablility of obtaining the sum i}).$$
This will give your answer.
You don't need to manually calculate the probability of obtaining the sum $i$ for each $i$. Consider the set $$A_i=\{1,2,...,i-1\}.$$ Note that $i=1+(i-1)=2+(i-2)=\cdots.$
The couplets $(1,i-1),(i-1,1),(2,i-2),(i-2,2),...$ correspond to the sum $i.$ For even $i,$ we have $$2(\frac{i-1}2)=i-1$$ many couplets giving the sum $i$ and for odd $i,$ we have $$2(\frac i2 - 1)+1=i-1$$ many couplets giving the sum $i$. That means it doesn't matter whether $i$ is even or odd.