It goes up and down a lot, in a major way. Is it me not understanding the axes or are the flips a bit on the pseudo-non-random side? I would expect a very high probability against what appears to be a non-random distribution.
Well, I have transformed the data in a special way that changes its properties. The first way is to rescale the differences as a ratio of the standard deviation of the difference (this goes up with the square root of the number of flips). The second way is to log-transform the x-axis, because this makes the behaviour look similar for any piece of the x-axis. (I think I have that right). The net result is that you have graph which is centred on zero (net difference zero) and tends to stay fairly near (with the frequency of the numbers on the y-axis being just like a standard normal distribution).
An interesting fact is that when the difference crosses zero it is likely to cross zero several more times in the near future. When it gets far from zero it is likely to take quite a while to get back again. At any time the coins don't know what the total is so far, so, untransformed it is a random walk starting whereever it happens to be.
Thanks. Appreciate it. Messaged you.