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.

Bill, its no problem being tired, we've all been there.  Chess and go as the whole game don't have surrender options for the game itself but there are things you can do, be challenged on, and then give up to save time, material, or there's a better opportunity to gain space.  So in the small picture, you can realize there's not much value to be gain in continuing a certain course of actions.  In fact in Go you are supposed to stop playing once both people hit the point of no more territory you can expect to be gained.

WARK!

Thus Spoketh Winterclaw: a blog about a Winterclaw who speaks from time to time.

 

(using Poser Pro 2014 SR3, on 64 bit Win 7, poser units are inches.)