Here is a part of the text of the game theory book. I got confused because of probability, which is here in a red frame. I understood that to obtain it, we multiply expression from the middle with probability $\frac{1}{n}$, but why we here should multiply? Are these probabilities independent? Also, I'm a bit lost with an interpretation of this "red" probability.
this is incomplete, but as no one else has tried... In these sort of game values, such as Shapley value, you get your value from the coalitions you are in. So if you are in a coalition, you can leave and ruin the coalition. So the first set of equations, if there is a coalition of size s, in a population of n, what are the chances you are in that coalition. Then 1/n shows up, because as the problem says, each size of coalition is equally likely,so each of those probabilities is multiplied by 1/n.