Given that I can see how much total hashing power is in runs, and I can see how much hashing power I am contributing to each run, how can I calculate an average rate of BTC I will get?
To go from GH/sec to BTC/day, use this:
BTC/day = GH/sec * 1,000 * 25 * 24 * 3,600 / 2^32 / difficulty(in millions)
Because: the block reward is currently 25, there are 24 hours in a day, 3600 seconds in an hour, and the base share difficulty is 2^32.
So 100GH/s, at the current difficulty of 707 million will yield:
100 * 1,000 * 25 * 24 * 3600 / 2^32 / 707.41 = .0711 BTC/day
Note that this assumes no pool fees and no variance at all. Of course, the difficulty is increasing so the yield will drop.