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Comment by orlp | Hacker News Reader

How to create a fair coin from an arbitrarily biased coin:

1. Toss the coin and remember the answer.

2. Toss the coin again, if it is different from your previous toss then your result from #1 is fair. Otherwise, go back to step 1.

If p is the probability of getting heads, there are four possible outcomes with their associated probabilities:

    TT -> (1 - p)^2   (rejected)
    HT -> p * (1 - p)
    TH -> (1 - p) * p
    TT -> p^2         (rejected)

Needless to say, p * (1 - p) and (1 - p) * p have an equal probability, so if we don't reject our two tosses, we have a fair outcome.

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Reference: https://en.wikipedia.org/wiki/Randomness_extractor#Von_Neuma...

VN extrator is a specific case of a more general idea: When you independently (hard assumption of VN extractor) draw M times with N possibilities then you can extract entropy from their permutation.

Assign some scheme for converting permutations to an index.

Then get uniform bits out, maintain two variables: one is the product of the number of permutations, the other gets multiplied by the number of permutations and the index added. Whenever the number of possibilities is divisible by two, output the LSB of the index accumulator and halve the number of possibilities.

Size up your groups and accumulators and you can get arbitrarily high extraction rates.

Doing it efficiently and in constant time (e.g. without divisions) is the more exciting trick. A colleague and I managed an extractor for the binary case that packs takes 10+3N multiplies and N CTZs to pack N bits (giving an exact invertible encoding when bits choose ones is < 2^64).

"arbitrarily" is doing some heavy lifting!

I'm not sure that two concurrent harmonious answers constitutes a "fixed" coin or a diagnosis of a fixed coin.

This scheme will be rubbish with a one sided coin ie the limit for "arbitrary fixed coin".

How is that "heavy lifting"? It's perfectly reasonable for any real-world "coin".

1. flip the coin until it lands on its edge.

2. the person who achieves this is the winner.

That's cute. intuitively, if two flips give different outcomes, it's fifty/fifty which would be first.

But also, you might have to flip the coin an arbitrarily large number of times before you get a "heads tails" or "tails heads" roll (if I can arbitrarily pick how biased the coin is).

The opening scene of "Rosencrantz and Guildenstern Are Dead" springs to mind.

And that coin wasn't even biased... although Tom Stoppard was a confounding factor.

You are assuming an unbiased coin.

Imagine I glue a poker chip to a washer. There's a clear bias in the outcome of this "coin".

This method resolves that bias.

I understood perfectly already.

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