Anytime I said "half a share" I am speaking in a probability sense, not a literal sense.
That's there problem. There is no "halfway" in the probability sense. That's exactly what the gambler's fallacy is all about. If you're throwing dice and trying to roll a six and the first three aren't six, it's incorrect to say "Well gee, it's a 1/6 chance and I've thrown it 3 times, so I must be halfway there!" The harsh reality is you've accomplished nothing. No amount of missed rolls bring you any closer to what you're after. You could roll one hundred non-sixes in a row and the 101st roll would still be a 1/6 chance; no more likely then the first roll. A die is just a piece of wood or plastic, the way physics behaves on it remains the same.
If we are speaking of probability before the events occur, then it is correct to say that over the course of 6 rolls, on average 3 will be heads and 3 will be tails.
But yes, once you actually start making the flips, you cannot make any determinations of the next flip, based on the previous ones.
Same with mining. One can calculate beforehand, the average number of shares that they might submit for a given difficulty over a period of time, but the actual results will vary and have no bearing on previous results. A higher difficulty is going to to have a higher variance, and one can argue that a lower difficulty should be used to provide more "fair" (or even) payouts over the short term. But if you are mining for weeks/months, it should all even out. Some days you will be unlucky and get less than you "should", other days you will be lucky and get more than you "should".
The whole point of difficulty is to let the pool know how much you are working. Ideally you would submit every single *hash* that you produce to the pool. In that case the pool's hashrate would exactly match your hashrate, and you would be paid perfectly based on your percentage of work. But that's obviously completely impractical. With a higher difficulty, you submit shares less often, which will cause some variance between your hashrate, and the hashrate that the pool assumes you are producing. The tradeoff between difficulties is reasonable to debate, but it boils down to how long a period of time is required to even out the variance.
I personally believe that the current difficulty is too high for low hashers to receive a reasonable variance, but that's just my opinion.
