Motivation: A friend asked me this question.
The Problem: Suppose you start off with a dollar. You flip a fair coin, if it lands on heads you win $50$ cents otherwise you lose $50$ cents. If after $n$ flips you have a nonzero amount of money, you win. What's the probability you win? What about the limiting case as $n$ tends to infinity?
edit: In this game you are not allowed to have negative money. Thanks, Jonathan Fischoff, the linked helped greatly.