Quote - The law of averages is the common term for the gambler's fallacy. 

Given time and a large enough sample, switching will approach but not reach (or maintain) exactly 2/3rds.  However, as they say, the die doesn't know that it's due.

I read a 19th century book (whose title and writer I have forgotten about over the intervening 30 years) about mathematical crackpots (and other subjects), which described how the author and George Boole conducted experiments in probability. Back in the early 19th century, it was not at all clear whether our current notion (that successive flips of a fair coin, or rolls of a fair die, are statistically unrelated) was correct. They assumed this as the null hypothosis, but also wanted to investigate whether what we now know as the "gambler's fallacy" was true (that an imbalance of heads would be counteracted by an imbalance of tails), or whether the notion of streaks was true (i.e., that after several heads in a row, the next flip would be more likely heads). So you had these mathemticians rolling dice and flipping coins and writing down the results. Pretty cool.