The general study of such things is called “randomness extractors”. The new Gödel prize goes to this paper which shows how to extract a nearly perfect random bit out of two sources with low min-entropy.
I realize this sounds like a minor detail to someone who finds this cool (and so do I), but I don't think it is. It's kind of frustrating to be told that your intuition is wrong by someone smarter than you, when your intuition is actually correct and the experts are moving the goalposts. IMO, it makes people lose respect for experts/authority.
I suppose we'll have to disagree about whether this is incredible. The response shows that (1) this can be done at all, and (2) that the answer is exponentially likely as time goes on, not asymptotically, but for finite n. Incredible! You don't see finite-decay bounds very often! If you don't think that's incredible I invite you to ask a room full of people, even with the qualifications you deem appropriate, e.g., "solution does not need to be constant-time", or whatever.
I'm sure it's possible to make a coin with what one might term "complex bias" where the bias extends over two events (thick gloop inside the coin, possibly heat activated or non-Newtonian).
This method sounds like the bias needs to be fixed ("simple bias" if you like)?
I guess that's just out of scope here.
Aside: There's a strong 'vibe' that HHHH HHHH with a single coin is somehow less random than HTHTHTHT HTHTHTHT with two coins when discarding HH and TT. I wonder what the origin of that feeling is - maybe just HT doesn't seem like it's a constant result simply due to nomenclature? If we call HT "type-A", then we just have HHHH HHHH vs. AAAA AAAA; and I _feel_ happier about the result!
HPMOR•3h ago