Math Question

I often play blackjack on this site to keep my game sharp. I've noticed that in the course of an hour of play you can go from being ahead to being behind several times.

So I was wondering: Say you were in a computer generated game of coin flipping with 60 decisions a minute. How many times could you expect to be ahead, and then find yourself behind in an hour? A day? A year?

I'm assuming that the longer you play the game the less likely the swings would be. The reason being that once you were ahead (or behind) say, 30 or so units, it would become less likely that you would win/lose that many units in order to find yourself at zero again. And after, say several thousand plays, you might find yourself several hundred units ahead or behind, making a reversal even less likely.

I'm sure that this idea had been written about exhaustively, somewhere, and I was wondering if anyone here could comment on this math question.