Reading the editorial of this atcoder problem agc028_b (editorial), I noticed that the solution was similar to the solution of D. Score of a Tree (editorial).

Both of those problems define a function to calculate a value based on a random initial configuration of elements. The problems then ask for the sum of values for every possible initial configuration. The editorial solutions involve calculating the expected value of the result and multiplying it by some other value (the contribution of each element or the amount of possible configurations).

My question is, how can we relate the expected value of a result to the exact sum of every possible result? Do you guys have other problems involving that technique or sources that explain it?

Thanks!