What are the characteristics of coin tossing systems which would prevent (or at least bias against) a fair coin from producing 100 heads in a row compared to any other sequence of heads or tails?
No, you're right, it can't be the one and do the other at the same time.
Realized that while replying.
I was thinking that if we had a system that showed human-scale random behavior -- that is, if we agreed it was random, or "fair" -- then by definition it would show roughly an even number of each of the two possible end-states at a sufficiently large scale of results (otherwise it's biased).
So if we were to record each incident of an end-state (say, heads gets a red dot and tails gets a blue dot) then as long as we have a large enough pool of results, we'll see about an equal mix of red and blue dots in an arrangement on a string.
We're essentially using the coin as a measuring device (since we're not deciding anything based on the toss) that we dip into the system, reset, and dip in again, like taking the pressure on a tire with a tire gauge.
If you're using something like Diaconis's device, your results-space will be a solid color if all coins go in with the same side up, and it will be an irregular mix if the coins' starting orientations are randomized.
You could put a switch on the Diaconis machine to change the result of the flip, and use that with all-heads or -tails starting positions to create regular patterns of results.
But for a system that makes fair tosses, once randomness became evident, you'd have an irregular pattern with roughly equal portions of red and blue.
(Which is what makes all-reds or -blues different from the entire large set of patterns that are irregular and roughly equal, not just different from any one of them.)
So how large do clusters of red and blue get in this system? How large could they possibly get?
Well, the largest cluster of red or blue dots must contain less than half the number of measurements (flips) because there's as much red as there is blue on our chart. And it can't be half, because that would be the results-space from a system that is regular (2 red and blue hemistrings) or biased (nearly equal hemistrings, maybe with some foam in between), not random.
The largest cluster also can't be 1, because that would make the system perfectly regular like a metronome, which we know it's not.
So the largest cluster in a FCTS should be somewhere above 1 and somewhere below half the number of flips required for randomness to become sustainably evident.
If randomness always emerges by 200 flips or less -- if you can always detect randomness by that point -- then I'd have to be right that runs of 100 are impossible in a FCTS.
Well, maybe that's ok if you have some way of knowing the range of randomness your system can produce, but coin-flipping is open-ended, a potentially infinite string.
It could be that the results-space of human coin flipping is like a 2-D version of our universe, with vast expanses of weak noise and spots of enormous clumping which are gigantic relative to the bits of stuff that make them up but extremely small and slender relative to the space they inhabit.
Entering the current universe at a random point, even billions of times, you're not likely to land on one of the truly big clumps.
So we can agree that our system's fair, but if the results-space really is infinite, then you can't say anything about the size of the biggest clump, since your small-scale randomness, as representative of the entire system as it may be, could be right up next to a Jupiter of results that you haven't quite gotten to yet.
Which brings it down to the question of whether your physical set-up produces that kind of results-space or not.
For example, if you put a direction detector in a turbulent chamber and used it to print a ribbon by coordinating its direction with a color wheel, you'd get a random variation of colors streaming from the machine.
At the same time, there'd be a limit to the possible length of time you could get one color, if you've created a situation of continual turbulence, and that limit would be determined by how you set the mechanism up.
Is it possible to build a coin-flipper that does something like that? Randomizes the results but limits streaks?
I don't know. I guess you could do it, because even though you set it up to be trending one way or another at any given time, if you did it right there might be no way for the flippers to be sure they were in a trend, or to know which direction it was going, or when it would change, since there's no large-scale regular pattern, and local turbulence could conceal weak larger-scale trends.
In other words, you could make it "fair to the participants" within a certain number of flips even though it was always biased in one way or other and does have limits which you could figure out if you played it a very long time (let's say, a billion years).
Maybe someday someone will discover some necessary "turbulence" in the physical setup of fair human coin tossing that does keep it from trending too long in one direction, but if anybody ever does, it certainly won't be me, so I have to accept the possibility of the Jupiter of streaks looming out there somewhere on our infinite string.
But even if we assume that "roughly" meant the same deviation from 50% as in Piggy's scenario, the formulas show that you're asking a different question - if it had been the same question, the formulas would have been the same in the limit case of "no deviation at all".
