A frequently-ignored fact of life. If X and Y are random variables, then
E(X+Y) = EX + EY.
Even if X and Y are not independent. (It also behaves properly with respect to multiplication by a constant, but that does not lead to surprises).
Many proofs in probability and computer science exploit this to give simple proofs of fairly profound facts. One example is the proof that the average number of fixed points of a permutation is 1; algebraists do it the hard way, probabilists have an easy proof.
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