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.

I have to run, so I will just post the explanation. This is a parity argument. If the 2nd card has the same odd or even parity as the first card, then the sum is even, otherwise it is odd.

If the number of even and odd cards in the urn are equal, then the probability of an even number is 1/2. Otherwise, the probability of matching parities is greater than 1/2. Because the starting number of cards (5555) is odd, the number of odd and even cards are necessarily unequal, so the probability that S is even is necessarily greater than 1/2.

This is also often the case when the number of initial cards is even.

Numeric proof later today, unless somebody else posts it first.