Intuition behind negative covariance of two card draws?

Question

As seen in this link, the covariance of two draws from a standard set of 52 (with the mapping of A, J, Q, K to 1, 11, 12 and 13 respectively) is negative. Is there an intuitive reason as to why it is so?

Answer 1

If I ask you to pick two numbers at random without replacement from the set $\{1, 2, \ldots, n\}$, does it not make intuitive sense that, because the numbers cannot be equal, their covariance must be negative?

Although in the case of the card deck there are multiple cards with a given point value, the fact that one card is chosen means there are fewer remaining cards with that same point value; thus, the same intuition applies.