I understand the algebraic proof of the linearity of expectation, but I cannot grasp the intuition, especially when the random variables are not independent. It is even counterintuitive as you cannot simply add the probabilities of dependent random variables, but it does not affect on the expected value.
Can anyone explain the intuition behind the algebraic formula? How to think about it?
This site https://brilliant.org/wiki/linearity-of-expectation/ was using the idea of total rainfall. How much rain do we expect to get over the next 3 days? It should be the amount we expect to get over the first day, plus the second, and then third day's expected values. Even though it's not independent (for instance a big storm can take more than 1 day to blow through), our system must be able to add up the individual expected values to get the larger expected value or the system isn't working right.