From our current "unlucky" status, how long would it take before we have statistically more than 50% probability to catch up with the expected rewards? I believe this question can be (formally) answered. I'm not really expecting it, but if someone is both able and willing to calculate or even approximate it, I would find it useful and appreciate it :-)
There will never be more than 50% probability of being lucky, just as there's never more than 50% of probability of being unlucky.
Past luck doesn't affect future luck. It doesn't matter if the pool was lucky or unlucky recently; the probability of finding a block after
n times difficulty shares is always the same.
While there's certainly a possibility of evening out the past bad luck or even surpassing the expected rewards in the long haul, this
cannot be the likely (p > 0.5) scenario.
My inability to find a simple explanation of why actual payouts have been below expected caused me to move most of my pool work elsewhere.
Luck is just that: luck. There's no explanation for it. You either have bad luck or you don't.
Taking the last 100, 250 and 500 blocks from the statistics, I found that the mean of the required shares to solve a block was 1.216005475, 1.0990548393 and 1.0534418012 times difficulty, respectively.
Assuming a normal distribution and estimating the standard error from the respective samples (0.1262058917, 0.0744550916 and 0.0485402741 respectively), there's a chance of 4.36%, 9.18% and 13.57%, respectively, that a given pool would be that unlucky.
4.36% sounds pretty low, but since there are more than 23 mining pools, chances are that it happens to one of them...