You face an urn with 5555 cards in it, each has a non-zero integer written on it. Nothing is said about the distribution of those numbers. You are told to draw randomly a card, copy the number, return it back, shuffle and draw randomly a card, then write down the sum of both numbers, say S.

(i) Prove: The probability of S being an even number is higher than S being odd.

(ii) Is it true for any initial number of cards? Comment.

This problem is related to

Fair Pairs. The solution to that problem shows that if the initial number of cards is N^2 and the distribution has (N^2+N)/2 cards of one parity and (N^2-N)/2 cards of the opposing parity then the probability of drawing two cards of the same parity is 1/2.