The question is suppose we have a biased coin which lands head with probability $p = 0.99$. How can we find an efficient way to turn that into a fair coin (with less than 100 expected flips).
So in this type of question, the usual way when $p$ is not too different than 0.5 is to implement Von Neumann's Unfair Coin Solution. For example, we flip twice and assign "Head Tail" to Head and "Tail Head" to Tail.
However, as now the coin almost lands head and it will take a lot flips to stop the simulation. I am wondering if there is any algorithm that can improve the efficiency?
$(0.99)^n = 0.5\\ n\log(0.99) = \log 0.5\\ n = \frac {\log(0.5)}{\log(0.99)} \approx 69$
If you flip this coin 69 times with probability $0.4998$ you will see all heads. I can't see a way to do this with fewer than 69 flips.