Von Neumann Biased Coin Simulation

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Conditioning / Information Extraction.

Your Task
You are given a biased coin with an unknown probability $p \in (0, 1)$ of landing Heads ($H$), and $q = 1-p$ of landing Tails ($T$). Formulate an algorithm (the von Neumann extractor) that simulates an exact fair coin (probability $1/2$ for outcome 0 and $1/2$ for outcome 1) using independent tosses of this biased coin. Determine the expected number of biased tosses required to generate one fair bit.
Constraints

• $p \in (0, 1)$, $p$ is unknown and constant across tosses.