Red vs black

You have a regular deck of 52 playing cards which you shuffle very well. You are instructed to peel off two cards at a time and place them in one of three piles:

• Put both cards in the right pile if both cards are red.
• Put both cards in the left pile if both cards are black.
• Put both card in the middle pile if one is red and one is black.

You do this until you have run through all 52 cards. What is the probability that the number of cards in the right pile will be equal to the number of cards in the left pile?

Variation

What if you start with a deck that has 4 more black cards than red cards. How will the piles end up?

Note

This makes (in fact, originally was) a great magic trick: make a prediction, then run through the cards.