You have a deck of $52$ cards, and you keep taking pairs of cards out of the deck.
If a pair of cards are both red, then you win that pair; if a pair of cards are both black, then I win that pair; if a pair of cards has one red and one black, then it's discarded. If, after going through the whole deck, you have more pairs than I do, then you win $1$ dollar, and if I have more pairs than you or we have equal number of pairs do, I win $1$ dollar.
What is the value of this game in the long run?
So obviously the probability that either one of us has more pairs is the same, as there is symmetry. However, this game is rigged against me as if we have an equal number of pairs, then I still lose.
How do we calculate this probability, of having an equal number of pairs?
After the mixed pairs are discarded, there are an equal number of black and red still in the deck, so both players will have the same number of pairs.