Here is the simplest way to answer the question.

Let the random variable X1 = 1 if an ace turns up on the first card, 0 otherwise. E(X1) = 4/52 = 1/13. Likewise let X2 =1 when an ace turns up on the second card and E(X2) = 1/13. The expected number of aces is E(X1 + X2) = E(X1) + E(X2) = 1/13 + 1/13 = 2/13. The X1 and X2 are dependent random variables but when summing expected values it does not matter.