The insight here is that you need to emit two signals that has equal probability, even if those two signals are rare in the full distribution. In the full distr

The insight here is that you need to emit two signals that has equal probability, even if those two signals are rare in the full distribution. In the full distribution, you’re allowed to add any other kinds of signals that aren’t those two. You then throw out all signals that are not those two signals, and the conditional distribution will renormalise itself to give you a fair coin toss. You pay for this by throwing out many bits that are not these two signals. The less fair the coin, the more coin flips you throw away. In the trivial case of a fair coin, you throw away nothing and keep every coin toss. In a biased coin, you throw away any pairs of HH or TT. Independence is a major assumption underlying any of these models.