Riddle me this

[matthewdaly]

Huh. I just saw a puzzle that I've never seen before. And it's one of those gems that makes you say "THIS is why combinatorics is so cool; learn how to count around the thing that is hard to calculate." Actually, while writing this I just thought of an even cooler proof than my original proof, which was pretty cool in its own right.

Morpheus and Neo each have a pile of fair coins. Morpheus has x coins, Neo has x+1. They flip all of their coins. What is the probability that Neo has more coins heads up than Morpheus?