Biased Coin, Fair Game

What this probability-based coin simulation question tests

This is a medium-difficulty probability question that asks you to reason about von Neumann's classic procedure for extracting fair randomness from a biased source. It rewards clear setup and algebraic manipulation over numerical computation.

To solve it, you identify all four possible outcomes of two flips, assign probabilities based on the bias parameter, and determine which outcomes trigger acceptance versus rejection. The key insight is recognizing the symmetry that makes certain pairs of outcomes equally likely, even when the coin is biased. You then sum the relevant probabilities and simplify the expression.

• Independence and multiplication rule for sequential events
• Partitioning outcome space and exhaustive enumeration
• Algebraic simplification of probability expressions