E[aX + bY] = a E[X] + b E[Y], with no independence assumption anywhere.
That is the whole point, and it is why linearity solves problems that look intractable. To find the expected number of something, write it as a sum of indicator variables and add up their individual probabilities - the dependence between them never enters.
The trap. It does not extend to products (E[XY] = E[X]E[Y] needs independence) or to variance (Var(X+Y) = Var(X)+Var(Y) needs zero correlation). Candidates who internalise "no independence needed" often leak it into variance, which interviewers test deliberately.