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.