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.

(In reply to

A related problem which resolves part 2. by Brian Smith)

It is a different problem:

- the second draw is after returning 1st card

- nothing is known about the distribution of the numbers