I'm stuck on a very simple probability question that somehow isn't making sense to me: What is the expected number of face cards in 3 draws from a 52 card deck(without replacement)?
I understand that the answer here is 9/13(and that you can calculate it by summing the 0, 1, 2, and 3 card events * their weight). This also makes sense intuitively.
The answer described, however, uses indicator random variables for each draw(ie expected number of face cards on draw Xi). The answer simply calculates 3*12/52 to get 9/13. This approach, however, doesn't make sense to me - since card draws are dependent events, wouldn't the expected number of face cards on draw Xi differ from draw Xi-1? Even though the math lines up, what is the intuition behind this? If you used this approach for the with replacement case, the answer would be the same - this doesn't line up with the initial approach, however, and doesn't make sense intuitively either(you'd expect more face cards in 3 draws with replacement right?) Any insights on this?