"A "law of large numbers" is one of several theorems expressing the idea that as the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero. "
http://mathworld.wolfram.com/LawofLargeNumbers.html
And as far as what we're comparing in the fgators thread it's proportion vs. actual - but just consider that like number of heads flipped vs. # trials * expected # of heads - which does converge