This is pleasingly insane, congratulations! Is there a program to test the fairness of a given dice or coin? Is that a program that's even feasible to write?
You've always got the standard way to get fair random numbers from a fairness-unknown coin. Flip it twice. Restart if you get both heads or both tails. If you get H then T or T then H, those are equally probable, so take the first one of those as the final outcome.
This generalizes to a die of N sides. Roll it N times. If you don't get all N distinct results, restart. If you do, then take the first result as your final outcome.
(That may take a lot of trials for large N. It can be broken down by prime factorization, like roll 2-sided and 3-sided objects separately, and combine them for a d6 result.)
The key assumption is that T and H may not have the same probability, but each flip isn't correlated with past or future flips. Therefore, TH and HT have the same probability. So you can think of TH as "A" and HT as "B" then you repeatedly flip twice until you get one of those outcomes. So now your coin outputs A and B with equal probability.
I feel like I am missing something so obvious that I feel the need to correct wiki, but that likely means I am fundamentally missing the point.
"The Von Neumann extractor can be shown to produce a uniform output even if the distribution of input bits is not uniform so long as each bit has the same probability of being "one"->[first] and there is no correlation between successive bits.[7]"
As long as the person doesn't favor which of the two bits they chose is "first", then it should appear as random.
But that is self-defeating, as if the person had the capability to unbiased-ly choose between two binaries, they wouldn't need the coin.
But since the only way to determine the variation from expectation is repeatedly increasing sample size, I don't see how doing it twice, and just taking encoding of the bits, then...
Is the magic in the XOR step? To eliminate the most obvious bias (1v5 coin), until all that could had been left was incidental? Then, always taking the first bit, to avoid the prior/a priori requisite of not having a fair coin/choosing between two options?
and it clicked. Rubber duck debugging, chain of thought, etc.
It may be more likely that H or T happens (an unfair coin), but in a pair of H and T, both HT and TH are equally likely. Therefore which is "first" is equally likely H or T.
Only holds if no spooky effects change results based on last result. (like a magic die that counts upwards or a magic coin that flips T after H no matter what)
Your second paragraph is correct and may be where the previous poster's intuition was disagreeing, that the method doesn't necessarily hold for repeated iterations in a physical system where one trial starts from where the last one ended.
It's not even really "spooky" - all you need is a flipping apparatus that's biased towards an odd number of rotations, and so then THTH is more common than THHT and you get a bias towards repeating your last result.
I suspect that when the user is loading coins or dice in the machine, they would notice any dirt that was significant enough to look as though it might be a problem.
And oil deposits from your fingerprints I would imagine are so minuscule as to be insignificant in creating varying bias.
Even then, in both cases, you could wipe the objects with an alcohol swab before putting them into the shaker cups.
It could be argued, I suppose, that every micro-collision of the coin or die with the cup removes a few atoms, but I would suggest that its effect on the bias of the coin or die over time is again minuscule. Indeed, unmeasurable over a full sequence of cycles (128 for example) of the machine when generating a Bitcoin key.
I love the slow pace of the video, including a few minutes presentation of all available programs. And indeed, there are programs to test dice and coin bias:
....you can do that using our universe's physical constants too.
Care to elaborate? Or link?
I mean, everything that is, is just displaced temporarily homogeneous complexity, allowable between the fluctuations of gradients of temperature, allowing the illusion of work to appear more than just energy dissipating into the either of expanding space-time, dragged by the irreconcilability idea of "gravity".
But that doesn't help bake an Apple pie from scratch, as Carl Sagan would put it.
This generalizes to a die of N sides. Roll it N times. If you don't get all N distinct results, restart. If you do, then take the first result as your final outcome.
(That may take a lot of trials for large N. It can be broken down by prime factorization, like roll 2-sided and 3-sided objects separately, and combine them for a d6 result.)